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width: 0.3em;"></span> <a href="/nlab/show/diff/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/1155/#Item_89" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #33 to #34: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a><ul><li><a href='#transgression_of_differential_forms'>Transgression of differential forms</a></li><li><a href='#by_pullback_and_fiber_integration'>By pullback and fiber integration</a></li><li><a href='#transgression_of_forms_through_fiber_bundles'>Transgression of forms through fiber bundles</a></li></ul></li><li><a href='#examples'>Examples</a><ul><li><a href='#transgression_in_de_rham_cohomology'>Transgression in de Rham Cohomology</a></li><li><a href='#transgression_of_de_rham_cohomology_to_loop_spaces'>Transgression of de Rham cohomology to loop spaces</a></li><li><a href='#transgression_in_group_cohomology'>Transgression in group cohomology</a></li><li><a href='#InternalHom'>Transgression to mapping spaces in terms of internal homs</a></li></ul></li><li><a href='#related_entries'>Related entries</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>Transgression is a tool in <a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a> to transfer <a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a> classes from one <a class='existingWikiWord' href='/nlab/show/diff/space'>space</a> to another without needing a <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphism</a> between them. Rather, one has a third space with morphisms to each of the two spaces. Of course, not just <em>any</em> third space will do.</p> <p>The topological set up is as follows. We have two <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>s, <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>, and a <a class='existingWikiWord' href='/nlab/show/diff/generalized+cohomology'>generalised cohomology theory</a> <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>E^*(-)</annotation></semantics></math>. We want to be able to transfer ( <em>transgress</em> ) <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>E^*</annotation></semantics></math>-classes from <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>. We find a third space, <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Z</mi></mrow><annotation encoding='application/x-tex'>Z</annotation></semantics></math>, which has morphisms to both <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>. We usually write this in the following way:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable displaystyle='false' rowspacing='0.5ex'><mtr><mtd><mi>Z</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin{matrix} Z & \to& X \\ \downarrow \\ Y \end{matrix} </annotation></semantics></math></div> <p>(Note to self: replace with SVG, write <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>Z</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>f \colon Z \to X</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>Z</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>g \colon Z \to Y</annotation></semantics></math>.)</p> <p>An important example is that where we have some parameter space <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Σ</mi></mrow><annotation encoding='application/x-tex'>\Sigma</annotation></semantics></math> (for instance the circle <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Σ</mi><mo>=</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>\Sigma = S^1</annotation></semantics></math>) and set <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Z</mi><mo>=</mo><mi>Σ</mi><mo>×</mo><mo stretchy='false'>[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>Z = \Sigma \times [\Sigma,X]</annotation></semantics></math>, the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a> of <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Σ</mi></mrow><annotation encoding='application/x-tex'>\Sigma</annotation></semantics></math> with the <a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>mapping space</a> <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[\Sigma,X]</annotation></semantics></math> of maps from <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Σ</mi></mrow><annotation encoding='application/x-tex'>\Sigma</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> (for instance the <a class='existingWikiWord' href='/nlab/show/diff/loop+space'>loop space</a> of <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>). Then the morphism on the right is taken to be evaluation and the morphism on the left is projection onto the mapping space <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[\Sigma,X]</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>Σ</mi><mo>×</mo><mo stretchy='false'>[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy='false'>]</mo></mtd> <mtd><mover><mo>→</mo><mi>ev</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo stretchy='false'>[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy='false'>]</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ \Sigma \times [\Sigma, X] &\stackrel{ev}{\to}& X \\ \downarrow^{\mathrlap{p_2}} \\ [\Sigma,X] } \,. </annotation></semantics></math></div> <p>This setup induces <strong>transgression to mapping spaces</strong> in the following.</p> <p>Now we apply the generalised cohomology theory to this diagram. As it is a cohomology theory, the arrows reverse. We have part of our route from <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>E^*(X)</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>E^*(Y)</annotation></semantics></math>, namely from <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>E^*(X)</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>Z</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>E^*(Z)</annotation></semantics></math>. However, the way from <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>E^*(Y)</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>Z</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>E^*(Z)</annotation></semantics></math> is blocked: it is a one-way street and we want to go the <em>wrong</em> way.</p> <p>However, under certain special circumstances, cohomology theories admit push-forward maps. That is, if <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>Z</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>g \colon Z \to Y</annotation></semantics></math> is particularly nice, there is a map <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>g</mi> <mo>!</mo></msub><mo lspace='verythinmathspace'>:</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>Z</mi><mo stretchy='false'>)</mo><mo>→</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>g_! \colon E^*(Z) \to E^*(Y)</annotation></semantics></math>. The notation <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>g</mi> <mo>!</mo></msub></mrow><annotation encoding='application/x-tex'>g_!</annotation></semantics></math> (pronounced “g shriek”) has the explanation that it is a surprise that there is a map in this direction<sup id='fnref:1'><a href='#fn:1' rel='footnote'>1</a></sup>. Note that the push-forward map, <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>g</mi> <mo>!</mo></msub><mo lspace='verythinmathspace'>:</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>Z</mi><mo stretchy='false'>)</mo><mo>→</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>g_! \colon E^*(Z) \to E^*(Y)</annotation></semantics></math>, often results in a change of degree. This is described at <a class='existingWikiWord' href='/nlab/show/diff/fiber+integration'>fiber integration</a> and <a class='existingWikiWord' href='/nlab/show/diff/Pontrjagin-Thom+collapse+map'>Pontrjagin-Thom collapse map</a>.</p> <p>Thus the key to transgression is to understand the conditions where the map <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>Z</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>g \colon Z \to Y</annotation></semantics></math> admits a push-forward map. The simplest case is where <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Z</mi></mrow><annotation encoding='application/x-tex'>Z</annotation></semantics></math> is a product space, <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo>×</mo><mi>W</mi></mrow><annotation encoding='application/x-tex'>Y \times W</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi></mrow><annotation encoding='application/x-tex'>W</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/orientation'>orientable</a> for the generalised cohomology theory <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>E^*(-)</annotation></semantics></math>. This means that <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi></mrow><annotation encoding='application/x-tex'>W</annotation></semantics></math> has a fundamental class in <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>W</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>E^*(W)</annotation></semantics></math> and evaluation on this class gives a morphism <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>Y</mi><mo>×</mo><mi>W</mi><mo stretchy='false'>)</mo><mo>→</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>E^*(Y \times W) \to E^*(Y)</annotation></semantics></math>.</p> <h2 id='definition'>Definition</h2> <h3 id='transgression_of_differential_forms'>Transgression of differential forms</h3> <p>See at <em><a class='existingWikiWord' href='/nlab/show/diff/transgression+of+differential+forms'>transgression of differential forms</a></em>,</p> <h3 id='by_pullback_and_fiber_integration'>By pullback and fiber integration</h3> <p>(…)</p> <h3 id='transgression_of_forms_through_fiber_bundles'>Transgression of forms through fiber bundles</h3> <p>A related construction called <em>transgression</em> is the following</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>P \to X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/fiber+bundle'>fiber bundle</a> with typical fiber <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>:</mo><mi>F</mi><mo>↪</mo><mi>P</mi></mrow><annotation encoding='application/x-tex'>i : F \hookrightarrow P </annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>ω</mi><mo stretchy='false'>]</mo><mo>∈</mo><msubsup><mi>H</mi> <mi>dR</mi> <mi>n</mi></msubsup><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>[\omega] \in H_{dR}^n(F)</annotation></semantics></math> a class in <a class='existingWikiWord' href='/nlab/show/diff/de+Rham+complex'>de Rham cohomology</a> of the fiber, we say this is <em>transgressive</em> if</p> <ul> <li> <p>there exists a form <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>cs</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy='false'>(</mo><mi>P</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>cs \in \Omega^{n}(P)</annotation></semantics></math> on the total space of the bundle;</p> </li> <li> <p>such that restricted along <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>↪</mo><mi>P</mi></mrow><annotation encoding='application/x-tex'>F \hookrightarrow P</annotation></semantics></math> to the fiber it is <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∼</mo><mi>ω</mi></mrow><annotation encoding='application/x-tex'>\sim \omega</annotation></semantics></math>;</p> </li> <li> <p>and such that <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mi>cs</mi></mrow><annotation encoding='application/x-tex'>d cs</annotation></semantics></math> is the pull-back of a form <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>κ</mi><mo>∈</mo><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\kappa \in \Omega^{n+1}(X)</annotation></semantics></math> on the base along the bundle projection .</p> </li> </ul> <p>See for instance section 9 of (<a href='#Borel'>Borel</a>).</p> <p>This construction really exhibits transgression as a special case of the <a class='existingWikiWord' href='/nlab/show/diff/connecting+homomorphism'>connecting homomorphism</a> in cohomology</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>)</mo><mo>→</mo><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> H^n(F) \to H^{n+1}(X) </annotation></semantics></math></div> <p>that is induced from the <a class='existingWikiWord' href='/nlab/show/diff/exact+sequence'>short exact sequence</a> of cocycles</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ker</mi><mo stretchy='false'>(</mo><msup><mi>i</mi> <mo>*</mo></msup><mo stretchy='false'>)</mo><mo>↪</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>P</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mrow><msup><mi>i</mi> <mo>*</mo></msup></mrow></mover><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> ker(i^*) \hookrightarrow \Omega^\bullet(P) \stackrel{i^*}{\to} \Omega^\bullet(F) </annotation></semantics></math></div> <p>by restricting to those elements for which it factors through</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>↪</mo><mi>ker</mi><mo stretchy='false'>(</mo><msup><mi>i</mi> <mo>*</mo></msup><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \Omega^\bullet(X) \hookrightarrow ker(i^*) \,. </annotation></semantics></math></div> <p>For more on this see <a class='existingWikiWord' href='/nlab/show/diff/Chern-Simons+element'>Chern-Simons element</a>.</p> <h2 id='examples'>Examples</h2> <h3 id='transgression_in_de_rham_cohomology'>Transgression in de Rham Cohomology</h3> <p>With a <a class='existingWikiWord' href='/nlab/show/diff/generalized+cohomology'>cohomology theory</a> that has a good geometric model, such as <a class='existingWikiWord' href='/nlab/show/diff/de+Rham+complex'>de Rham cohomology</a>, there is often a similar geometric model for the push-forward map. In the case of de Rham cohomology, it goes by the name of <em><a class='existingWikiWord' href='/nlab/show/diff/fiber+integration'>integration along the fibres</a></em>.</p> <p>Using the notation above, with the simple case of <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Z</mi><mo>=</mo><mi>Y</mi><mo>×</mo><mi>W</mi></mrow><annotation encoding='application/x-tex'>Z = Y \times W</annotation></semantics></math>, we have the formula</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo>∈</mo><msubsup><mi>H</mi> <mi>dR</mi> <mi>k</mi></msubsup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>↦</mo><msub><mo>∫</mo> <mi>W</mi></msub><msup><mi>f</mi> <mo>*</mo></msup><mi>α</mi><mo>∈</mo><msubsup><mi>H</mi> <mi>dR</mi> <mrow><mi>k</mi><mo>−</mo><mi>dim</mi><mi>W</mi></mrow></msubsup><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \alpha \in H^k_{dR}(X) \mapsto \int_W f^* \alpha \in H^{k - \dim W}_{dR}(Y) </annotation></semantics></math></div> <h3 id='transgression_of_de_rham_cohomology_to_loop_spaces'>Transgression of de Rham cohomology to loop spaces</h3> <p>A particular application of this is in respect to the cohomology of <a class='existingWikiWord' href='/nlab/show/diff/loop+space'>loop spaces</a>, and in particular to <a class='existingWikiWord' href='/nlab/show/diff/iterated+integral'>iterated integrals</a>. The aim here is to transfer information from a space <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> to its <a class='existingWikiWord' href='/nlab/show/diff/free+loop+space+object'>free loop space</a>, <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>L X</annotation></semantics></math>. The intermediate space in this situation is <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><mi>L</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>S^1 \times L X</annotation></semantics></math> with evaluation as the map to <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> and projection to <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>L X</annotation></semantics></math>. Note that although <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>L X</annotation></semantics></math> is rather large, the <a class='existingWikiWord' href='/nlab/show/diff/fiber'>fibre</a> is <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>S^1</annotation></semantics></math> and it is that which needs to be controlled.</p> <p>If we worked in the realm of pure <a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a> (i.e. didn’t care about geometry) this transfer would be almost trivial. Indeed, if we worked with <em>based</em> loops, it would be a tautology since then the push-forward <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>H</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>∧</mo><mi>Ω</mi><mi>X</mi><mo stretchy='false'>)</mo><mo>→</mo><msup><mi>H</mi> <mrow><mo>*</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>Ω</mi><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H^*(S^1 \wedge \Omega X) \to H^{*-1}(\Omega X)</annotation></semantics></math> is simply the suspension isomorphism. (We use the <a class='existingWikiWord' href='/nlab/show/diff/smash+product'>smash product</a> as we are working with <em>based</em> spaces here.)</p> <p>However, we wish to have a geometric interpretation of this, and so work in the realm of <a class='existingWikiWord' href='/nlab/show/diff/differential+topology'>differential topology</a>: i.e. with smooth <a class='existingWikiWord' href='/nlab/show/diff/manifold'>manifold</a>s (albeit possibly infinite dimensional). We therefore have the diagram:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><mi>L</mi><mi>M</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>M</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mi>L</mi><mi>M</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ S^1 \times L M & \to & M \\ \downarrow \\ L M } </annotation></semantics></math></div> <p>The general lore of transgression says that this induces a map in cohomology <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>H</mi> <mi>dR</mi> <mi>k</mi></msubsup><mo stretchy='false'>(</mo><mi>M</mi><mo stretchy='false'>)</mo><mo>→</mo><msubsup><mi>H</mi> <mi>dR</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy='false'>(</mo><mi>L</mi><mi>M</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H^k_{dR}(M) \to H^{k-1}_{dR}(L M)</annotation></semantics></math> via</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo>↦</mo><msub><mo>∫</mo> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub><msup><mi>ev</mi> <mo>*</mo></msup><mi>α</mi><mo>.</mo></mrow><annotation encoding='application/x-tex'> \alpha \mapsto \int_{S^1} ev^*\alpha. </annotation></semantics></math></div> <p>As with any cohomological construction, there is always the question as to whether or not it is purely cohomological or whether or not it can be lifted to whatever-it-was that defined the cohomology theory. In this case, that is <a class='existingWikiWord' href='/nlab/show/diff/differential+form'>differential form</a>s. In this case we are in the happy circumstance that the lift exists. Expressed purely in terms of differential forms, the formula doesn’t change. However, we can think of differential forms as “that which acts on <a class='existingWikiWord' href='/nlab/show/diff/tangent+bundle'>tangent vector</a>s” and ask for a formulation that involves tangent vectors. This can make the integration step clearer.</p> <p>To reformulate transgression thus we need to understand tangent vectors on <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>L M</annotation></semantics></math>. The simplest way to view a tangent vector on <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>L M</annotation></semantics></math> is to identify <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi><mi>L</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>T L M</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mi>T</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>L T M</annotation></semantics></math>. That is, we view a tangent vector <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>T</mi> <mi>γ</mi></msub><mi>L</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>x \in T_\gamma L M</annotation></semantics></math> as a loop in <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>T M</annotation></semantics></math> with the property that <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo>∈</mo><msub><mi>T</mi> <mrow><mi>γ</mi><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo></mrow></msub><mi>M</mi></mrow><annotation encoding='application/x-tex'>x(t) \in T_{\gamma(t)} M</annotation></semantics></math>. In one view (<math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi><mi>L</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>T L M</annotation></semantics></math>), <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> tells <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi></mrow><annotation encoding='application/x-tex'>\gamma</annotation></semantics></math> which direction to move in; in the other view (<math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mi>T</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>L T M</annotation></semantics></math>), each <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>x(t)</annotation></semantics></math> tells each <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\gamma(t)</annotation></semantics></math> which direction to move in.</p> <p>That done, we have an obvious evaluation map <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><mi>T</mi><mi>L</mi><mi>M</mi><mo>→</mo><mi>T</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>S^1 \times T L M \to T M</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>↦</mo><mi>x</mi><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>x \mapsto x(t)</annotation></semantics></math> and so, given a differential form on <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math>, we can evaluate it on tangent vectors coming from <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>L M</annotation></semantics></math>.</p> <p>However, that’s not all we need. There’s a very special <a class='existingWikiWord' href='/nlab/show/diff/vector+field'>vector field</a> on <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>L M</annotation></semantics></math> which we want to throw into the mix. This is the vector field which assigns to <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi><mo>∈</mo><mi>L</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>\gamma \in L M</annotation></semantics></math> the tangent vector <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi><mo>′</mo><mo>∈</mo><mi>L</mi><mi>T</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>\gamma' \in L T M</annotation></semantics></math>. This is the rotation vector field: if we rotate the circle on <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><mi>L</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>S^1 \times L M</annotation></semantics></math> then to keep the evaluation map consistent, we have to rotate the loops in <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>L M</annotation></semantics></math> as well.</p> <p>So starting with a <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>-form, say <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi></mrow><annotation encoding='application/x-tex'>\alpha</annotation></semantics></math>, on <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math>, we fix a loop <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi><mo>∈</mo><mi>L</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>\gamma \in L M</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>t</mi><mo>∈</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>t \in S^1</annotation></semantics></math>, and take <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>k-1</annotation></semantics></math> tangent vectors at <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi></mrow><annotation encoding='application/x-tex'>\gamma</annotation></semantics></math>, say <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding='application/x-tex'>x_1, \dots, x_{k-1}</annotation></semantics></math>. We evaluate these at <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>t</mi></mrow><annotation encoding='application/x-tex'>t</annotation></semantics></math> to get <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo>∈</mo><msub><mi>T</mi> <mrow><mi>γ</mi><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo></mrow></msub><mi>M</mi></mrow><annotation encoding='application/x-tex'>x_1(t), \dots, x_{k-1}(t) \in T_{\gamma(t)} M</annotation></semantics></math> and also throw in <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\gamma'(t)</annotation></semantics></math>. That gives <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math> vectors at <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mrow><mi>γ</mi><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo></mrow></msub><mi>M</mi></mrow><annotation encoding='application/x-tex'>T_{\gamma(t)} M</annotation></semantics></math> on which we can evaluate <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi></mrow><annotation encoding='application/x-tex'>\alpha</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo stretchy='false'>(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>γ</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \alpha(x_1(t),x_2(t),\dots,x_{k-1}(t),\gamma'(t)) </annotation></semantics></math></div> <p>This is a number. Hence, by varying <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>t</mi><mo>∈</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>t \in S^1</annotation></semantics></math>, it defines a function <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>S^1 \to \mathbb{R}</annotation></semantics></math>. Integrating this function around <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>S^1</annotation></semantics></math> yields a number:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>∫</mo> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub><mi>α</mi><mo stretchy='false'>(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>γ</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mi>dt</mi></mrow><annotation encoding='application/x-tex'> \int_{S^1} \alpha(x_1(t),x_2(t),\dots,x_{k-1}(t),\gamma'(t)) dt </annotation></semantics></math></div> <p>Thus we have a map <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mo>⨂</mo> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>T</mi><mo stretchy='false'>(</mo><mi>L</mi><mi>M</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>\bigotimes^{k-1} T(L M) \to \mathbb{R}</annotation></semantics></math>. It can be shown that it is alternating and <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(k-1)</annotation></semantics></math>-linear, and that it varies smoothly over <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>L M</annotation></semantics></math>, hence defines an element of <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Ω</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>L</mi><mi>M</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Omega^{k-1}(L M)</annotation></semantics></math>.</p> <p>The reason why <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>∫</mo> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub><msup><mi>ev</mi> <mo>*</mo></msup><mi>α</mi></mrow><annotation encoding='application/x-tex'>\int_{S^1} ev^*\alpha</annotation></semantics></math> takes the form of the above formula is the following:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ev</mi> <mo>*</mo></msup><mo>:</mo><msup><mi>Ω</mi> <mi>k</mi></msup><mo stretchy='false'>(</mo><mi>M</mi><mo stretchy='false'>)</mo><mo>→</mo><msup><mi>Ω</mi> <mi>k</mi></msup><mo stretchy='false'>(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><mi>LM</mi><mo stretchy='false'>)</mo><mo>=</mo><msub><mo>⊕</mo> <mrow><mi>i</mi><mo>+</mo><mi>j</mi><mo>=</mo><mi>k</mi></mrow></msub><msup><mi>Ω</mi> <mi>i</mi></msup><mo stretchy='false'>(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy='false'>)</mo><mo>⊗</mo><msup><mi>Ω</mi> <mi>j</mi></msup><mo stretchy='false'>(</mo><mi>LM</mi><mo stretchy='false'>)</mo><mo>=</mo><msup><mi>Ω</mi> <mn>0</mn></msup><mo stretchy='false'>(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy='false'>)</mo><mo>⊗</mo><msup><mi>Ω</mi> <mi>k</mi></msup><mo stretchy='false'>(</mo><mi>LM</mi><mo stretchy='false'>)</mo><mo>⊕</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy='false'>(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy='false'>)</mo><mo>⊗</mo><msup><mi>Ω</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>LM</mi><mo stretchy='false'>)</mo><mo>.</mo></mrow><annotation encoding='application/x-tex'>ev^*: \Omega^k(M) \to \Omega^k(S^1\times LM)= \oplus_{i+j=k} \Omega^i(S^1) \otimes\Omega^j(LM) = \Omega^0(S^1) \otimes \Omega^k(LM) \oplus \Omega^1(S^1)\otimes \Omega^{k-1}(LM). </annotation></semantics></math></div> <p>Thus we may write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ev</mi> <mo>*</mo></msup><mi>α</mi><mo>=</mo><msub><mi>α</mi> <mn>0</mn></msub><mo>⊗</mo><msub><mi>α</mi> <mi>k</mi></msub><mo>+</mo><msub><mi>α</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>α</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo></mrow><annotation encoding='application/x-tex'>ev^*\alpha= \alpha_0\otimes \alpha_k + \alpha_{1}\otimes \alpha_{k-1}, </annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>α</mi> <mi>k</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mi>k</mi></msup><mo stretchy='false'>(</mo><mi>LM</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\alpha_k\in \Omega^k(LM)</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>α</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>∈</mo><msup><mi>Ω</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>LM</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\alpha^{k-1}\in \Omega^{k-1}(LM)</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>α</mi> <mn>1</mn></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy='false'>(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\alpha_1\in \Omega^1(S^1)</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>α</mi> <mn>0</mn></msup><mo>∈</mo><msup><mi>Ω</mi> <mn>0</mn></msup><mo stretchy='false'>(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\alpha^0\in \Omega^0(S^1)</annotation></semantics></math>. Then contracting with <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>S^1</annotation></semantics></math> via <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>∫</mo> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub></mrow><annotation encoding='application/x-tex'>\int_{S^1}</annotation></semantics></math>, only the second term survives. Thus</p> <div class='maruku-equation' id='eq:int'><span class='maruku-eq-number'>(1)</span><math class='maruku-mathml' display='block' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mo>∫</mo> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub><msup><mi>ev</mi> <mo>*</mo></msup><mi>α</mi><msub><mo stretchy='false'>)</mo> <mi>γ</mi></msub><mo stretchy='false'>(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>x</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>)</mo><mo>=</mo><msub><mo>∫</mo> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub><msub><mi>α</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>α</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>x</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>)</mo><mo>.</mo></mrow><annotation encoding='application/x-tex'> (\int_{S^1} ev^*\alpha)_{\gamma} (x_1, ..., x_{k-1}) =\int_{S^1} \alpha_{1} \otimes \alpha_{k-1} (x_1, ..., x_{k-1} ) . </annotation></semantics></math></div> <p>On the other hand, let <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>v</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo>∈</mo><msub><mi>T</mi> <mi>t</mi></msub><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><msub><mi>T</mi> <mi>γ</mi></msub><mi>LM</mi></mrow><annotation encoding='application/x-tex'>(v_i, x_i) \in T_t S^1 \times T_{\gamma}LM </annotation></semantics></math> be tangent vectors of <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi><mo stretchy='false'>(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><mi>LM</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>T(S^1\times LM)</annotation></semantics></math> for <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>k</mi></mrow><annotation encoding='application/x-tex'>i=1, \dots, k</annotation></semantics></math>. Then</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ev</mi> <mo>*</mo></msup><msub><mi>α</mi> <mrow><mo stretchy='false'>(</mo><mi>t</mi><mo>,</mo><mi>γ</mi><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>,</mo><mi>…</mi><mo>,</mo><mo stretchy='false'>(</mo><msub><mi>v</mi> <mi>k</mi></msub><mo>,</mo><msub><mi>x</mi> <mi>k</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>=</mo><msub><mi>α</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mo stretchy='false'>|</mo> <mi>γ</mi></msub><mo>⊗</mo><msub><mi>α</mi> <mn>1</mn></msub><msub><mo stretchy='false'>|</mo> <mi>t</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>,</mo><mi>…</mi><mo>,</mo><mo stretchy='false'>(</mo><msub><mi>v</mi> <mi>k</mi></msub><mo>,</mo><msub><mi>x</mi> <mi>k</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>+</mo><msub><mi>α</mi> <mi>k</mi></msub><msub><mo stretchy='false'>|</mo> <mi>γ</mi></msub><mo>⊗</mo><msub><mi>α</mi> <mn>0</mn></msub><msub><mo stretchy='false'>|</mo> <mi>t</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>,</mo><mi>…</mi><mo>,</mo><mo stretchy='false'>(</mo><msub><mi>v</mi> <mi>k</mi></msub><mo>,</mo><msub><mi>x</mi> <mi>k</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>.</mo></mrow><annotation encoding='application/x-tex'>ev^*\alpha_{(t, \gamma)}((v_1, x_1), \dots, (v_k, x_k)) = \alpha_{k-1}|_{\gamma} \otimes \alpha_1 |_t((v_1, x_1), \dots, (v_k, x_k)) + \alpha_k|_{\gamma}\otimes \alpha_0|_t((v_1, x_1), \dots, (v_k, x_k)).</annotation></semantics></math></div> <p>At the same time, <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>γ</mi></msub><mi>ev</mi><mo stretchy='false'>(</mo><mi>v</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>v</mi><mi>γ</mi><mo>′</mo><mo lspace='verythinmathspace' rspace='0em'>+</mo><mi>x</mi></mrow><annotation encoding='application/x-tex'>T_{\gamma} ev( v, x) = v \gamma' + x</annotation></semantics></math>. This can be seen by taking a variation (a small path) <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>t</mi><mo>+</mo><mi>v</mi><mi>ϵ</mi><mo>,</mo><msup><mi>γ</mi> <mi>ϵ</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(t+v\epsilon, \gamma^\epsilon)</annotation></semantics></math> representing <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>v</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(v, x)</annotation></semantics></math> (thus <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>γ</mi> <mn>0</mn></msup><mo>=</mo><mi>γ</mi></mrow><annotation encoding='application/x-tex'>\gamma^0=\gamma</annotation></semantics></math>). Then</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>γ</mi></msub><mi>ev</mi><mo stretchy='false'>(</mo><mi>v</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>ϵ</mi></mrow></mfrac><msub><mo stretchy='false'>|</mo> <mrow><mi>ϵ</mi><mo>=</mo><mn>0</mn></mrow></msub><mo stretchy='false'>(</mo><msup><mi>γ</mi> <mi>ϵ</mi></msup><mo stretchy='false'>(</mo><mi>t</mi><mo>+</mo><mi>v</mi><mi>ϵ</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>ϵ</mi></mrow></mfrac><msub><mo stretchy='false'>|</mo> <mrow><mi>ϵ</mi><mo>=</mo><mn>0</mn></mrow></msub><msup><mi>γ</mi> <mn>0</mn></msup><mo stretchy='false'>(</mo><mi>t</mi><mo>+</mo><mi>v</mi><mi>ϵ</mi><mo stretchy='false'>)</mo><mo>+</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>ϵ</mi></mrow></mfrac><msub><mo stretchy='false'>|</mo> <mrow><mi>ϵ</mi><mo>=</mo><mn>0</mn></mrow></msub><msup><mi>γ</mi> <mi>ϵ</mi></msup><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>v</mi><mi>γ</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo>+</mo><mi>x</mi><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo>,</mo></mrow><annotation encoding='application/x-tex'>T_{\gamma} ev( v, x)= \frac{d}{d\epsilon}|_{\epsilon=0}(\gamma^\epsilon(t+v\epsilon))=\frac{d}{d\epsilon}|_{\epsilon=0}\gamma^0(t+v\epsilon)+ \frac{d}{d\epsilon}|_{\epsilon=0}\gamma^\epsilon(t) = v\gamma'(t)+x(t),</annotation></semantics></math></div> <p>which is a tangent vector at <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\gamma(t)</annotation></semantics></math>. Thus</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ev</mi> <mo>*</mo></msup><mi>α</mi><msub><mo stretchy='false'>|</mo> <mrow><mo stretchy='false'>(</mo><mi>t</mi><mo>,</mo><mi>γ</mi><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>,</mo><mi>…</mi><mo>,</mo><mo stretchy='false'>(</mo><msub><mi>v</mi> <mi>k</mi></msub><mo>,</mo><msub><mi>x</mi> <mi>k</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>=</mo><msub><mi>α</mi> <mrow><mi>γ</mi><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><msub><mi>T</mi> <mi>γ</mi></msub><mi>ev</mi><mo stretchy='false'>(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>T</mi> <mi>γ</mi></msub><mi>ev</mi><mo stretchy='false'>(</mo><msub><mi>v</mi> <mi>k</mi></msub><mo>,</mo><msub><mi>x</mi> <mi>k</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>ev^*\alpha|_{(t, \gamma)}((v_1, x_1), \dots, (v_k, x_k))= \alpha_{\gamma(t)}(T_\gamma ev (v_1, x_1), \dots, T_\gamma ev (v_k, x_k)) </annotation></semantics></math></div><div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>=</mo><msub><mi>α</mi> <mrow><mi>γ</mi><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>k</mi></msub><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>+</mo><msub><mi>α</mi> <mrow><mi>γ</mi><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><msub><mi>v</mi> <mn>1</mn></msub><mi>γ</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>k</mi></msub><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>+</mo><mi>c</mi><mo>.</mo><mi>p</mi><mo>.</mo></mrow><annotation encoding='application/x-tex'> = \alpha_{\gamma(t)}(x_1(t), \dots, x_k(t)) + \alpha_{\gamma(t)}(v_1 \gamma'(t), x_2(t), \dots, x_{k}(t)) + c.p. </annotation></semantics></math></div> <p>Thus <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>α</mi> <mi>k</mi></msub><mo>=</mo><mi>α</mi></mrow><annotation encoding='application/x-tex'>\alpha_k=\alpha</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>α</mi> <mn>0</mn></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>\alpha_0=1</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>α</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>ι</mi><mo stretchy='false'>(</mo><mi>γ</mi><mo>′</mo><mo stretchy='false'>)</mo><mi>α</mi></mrow><annotation encoding='application/x-tex'>\alpha_{k-1}=\iota(\gamma')\alpha</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>α</mi> <mn>1</mn></msub><mo>=</mo><mi>dt</mi></mrow><annotation encoding='application/x-tex'>\alpha_1=dt</annotation></semantics></math>. Combining with equation <a class='maruku-eqref' href='#eq:int'>(1)</a>, we obtain the desired formula (might be up to a sign).</p> <p>There is much more to tell in this particular story. It is possible to replacing <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>S^1</annotation></semantics></math> by a <a class='existingWikiWord' href='/nlab/show/diff/simplex'>simplex</a> (but keeping <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_137' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>L M</annotation></semantics></math> the same) and so build up more complicated objects in <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_138' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Ω</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>L</mi><mi>M</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Omega^*(L M)</annotation></semantics></math>. This is the starting point of Chen’s theory of <a class='existingWikiWord' href='/nlab/show/diff/iterated+integral'>iterated integrals</a> and further developments of the theory can be found in the work of Jones, Getzler, and Petrack.</p> <h3 id='transgression_in_group_cohomology'>Transgression in group cohomology</h3> <p>See at <em><a class='existingWikiWord' href='/nlab/show/diff/transgression+in+group+cohomology'>transgression in group cohomology</a></em>.</p> <h3 id='InternalHom'>Transgression to mapping spaces in terms of internal homs</h3> <blockquote> <p>just a rough remark, for the moment</p> </blockquote> <p>At least for some cases of transgression to <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping space</a>s, the concept has an <a class='existingWikiWord' href='/nlab/show/diff/nPOV'>nPOV</a> description in terms of <a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>internal hom</a>s.</p> <p>This makes crucial use of the <a class='existingWikiWord' href='/nlab/show/diff/nPOV'>nPOV</a> notion of <a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a>, as described there.</p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_139' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding='application/x-tex'>\mathbf{H} = </annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_140' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>≃</mo></mrow><annotation encoding='application/x-tex'>\simeq</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/Infinity-Grpd'>∞Grpd</a>. For <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_141' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>X \in \mathbf{H}</annotation></semantics></math> and for <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_142' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Σ</mi></mrow><annotation encoding='application/x-tex'>\Sigma</annotation></semantics></math> a <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_143' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>-dimensional closed oriented <a class='existingWikiWord' href='/nlab/show/diff/manifold'>manifold</a>, and for <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_144' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> an <a class='existingWikiWord' href='/nlab/show/diff/abelian+group'>abelian group</a> that is injective as a <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_145' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Z}</annotation></semantics></math>-module, we may identify transgression to the mapping space <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_146' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[\Sigma,X]</annotation></semantics></math> as the composite</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_147' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>trans</mi> <mi>Σ</mi></msub><mo>:</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant='bold'><mi>B</mi></mstyle> <mi>n</mi></msup><mi>K</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mrow><mo stretchy='false'>[</mo><mi>Σ</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo></mrow></mover><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy='false'>]</mo><mo>,</mo><mo stretchy='false'>[</mo><mi>Σ</mi><mo>,</mo><msup><mstyle mathvariant='bold'><mi>B</mi></mstyle> <mi>n</mi></msup><mi>K</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mover><mo>→</mo><mrow><mo stretchy='false'>(</mo><msub><mi>τ</mi> <mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub><msub><mo stretchy='false'>)</mo> <mo>*</mo></msub></mrow></mover><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy='false'>]</mo><mo>,</mo><msub><mi>τ</mi> <mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub><mo stretchy='false'>[</mo><mi>Σ</mi><mo>,</mo><msup><mstyle mathvariant='bold'><mi>B</mi></mstyle> <mi>n</mi></msup><mi>K</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mo>≃</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy='false'>]</mo><mo>,</mo><msup><mstyle mathvariant='bold'><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><mi>K</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> trans_\Sigma : \mathbf{H}(X,\mathbf{B}^n K) \stackrel{[\Sigma,-]}{\to} \mathbf{H}([\Sigma,X], [\Sigma, \mathbf{B}^n K]) \stackrel{(\tau_{n-k})_*}{\to} \mathbf{H}([\Sigma,X], \tau_{n-k} [\Sigma, \mathbf{B}^n K]) \simeq \mathbf{H}([\Sigma,X], \mathbf{B}^{n-k} K) \,. </annotation></semantics></math></div> <p>Here the first step is application of the <a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>internal hom</a>, then the second is postcomposition with <a class='existingWikiWord' href='/nlab/show/diff/truncation'>truncation</a>, and then the last step uses the <a class='existingWikiWord' href='/nlab/show/diff/universal+coefficient+theorem'>universal coefficient theorem</a>. Note that the transgression map <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_148' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>trans</mi> <mi>Σ</mi></msub></mrow><annotation encoding='application/x-tex'>trans_\Sigma</annotation></semantics></math> depends on the choice of an equivalence <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_149' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>τ</mi> <mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub><mo stretchy='false'>[</mo><mi>Σ</mi><mo>,</mo><msup><mstyle mathvariant='bold'><mi>B</mi></mstyle> <mi>n</mi></msup><mi>K</mi><mo stretchy='false'>]</mo><mo>≃</mo><msup><mstyle mathvariant='bold'><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><mi>K</mi></mrow><annotation encoding='application/x-tex'>\tau_{n-k} [\Sigma, \mathbf{B}^n K]\simeq \mathbf{B}^{n-k} K</annotation></semantics></math>. Up to homotopy, this choice is precisely the datum of the <a class='existingWikiWord' href='/nlab/show/diff/orientation'>orientation</a> on <math class='maruku-mathml' display='inline' id='mathml_bbf456ef4c25aaf2805eefd365bd4f294a57b2f7_150' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Σ</mi></mrow><annotation encoding='application/x-tex'>\Sigma</annotation></semantics></math>.</p> <blockquote> <p>more details go here….</p> </blockquote> <p>This plays a role in the <a class='existingWikiWord' href='/nlab/show/diff/quantization'>quantization</a> process that yields <a class='existingWikiWord' href='/nlab/show/diff/functorial+field+theory'>FQFT</a>s. For an application see <a class='existingWikiWord' href='/nlab/show/diff/Dijkgraaf-Witten+theory'>Dijkgraaf-Witten theory</a>.</p> <blockquote> <p>more details, and more polishing later…</p> </blockquote> <h2 id='related_entries'>Related entries</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/transgression+of+bundle+gerbes'>transgression of bundle gerbes</a></li> </ul> <h2 id='references'>References</h2> <p>The classical notion of transgression of forms through fiber bundles is described in section 9 of</p> <ul> <li id='Borel'><a class='existingWikiWord' href='/nlab/show/diff/Armand+Borel'>Armand Borel</a>, <em>Topology of Lie groups and characteristic classes</em> Bull. Amer. Math. Soc. Volume 61, Number 5 (1955), 397-432. (<a href='http://projecteuclid.org/euclid.bams/1183520007'>EUCLID</a>)</li> </ul> <p>Transgression as <a class='existingWikiWord' href='/nlab/show/diff/pullback+in+cohomology'>pullback in cohomology</a> along the <a class='existingWikiWord' href='/nlab/show/diff/evaluation+map'>evaluation map</a> followed by <a class='existingWikiWord' href='/nlab/show/diff/fiber+integration'>fiber integration</a> over the base space is considered in:</p> <p>for <a class='existingWikiWord' href='/nlab/show/diff/differential+form'>differential forms</a>:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Jean-Luc+Brylinski'>Jean-Luc Brylinski</a>, §3.5 in <em>Loop Spaces, Characteristic Classes, and Geometric Quantization</em>, Birkhäuser (1993) [[doi:10.1007/978-0-8176-4731-5](https://doi.org/10.1007/978-0-8176-4731-5)]</li> </ul> <p>for <a class='existingWikiWord' href='/nlab/show/diff/Deligne+cohomology'>Deligne cohomology</a> (and with an eye towards defining the <a class='existingWikiWord' href='/nlab/show/diff/higher+parallel+transport'>higher holonomy</a> of <a class='existingWikiWord' href='/nlab/show/diff/connection+on+a+bundle+gerbe'>bundle gerbes with connection</a>):</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Ernesto+Lupercio'>Ernesto Lupercio</a>, <a class='existingWikiWord' href='/nlab/show/diff/Bernardo+Uribe'>Bernardo Uribe</a>, p. 2 of: <em>Holonomy for Gerbes over Orbifolds</em>, J. Geom.Phys. <strong>56</strong> (2006) 1534-1560 [[arXiv:math/0307114](https://arxiv.org/abs/math/0307114), <a href='https://doi.org/10.1016/j.geomphys.2005.08.006'>doi:10.1016/j.geomphys.2005.08.006</a>]</p> <blockquote> <p>(over <a class='existingWikiWord' href='/nlab/show/diff/orbifold'>orbifolds</a>)</p> </blockquote> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Christian+B%C3%A4r'>Christian Bär</a>, <a class='existingWikiWord' href='/nlab/show/diff/Christian+Becker'>Christian Becker</a>, Def. 9.1 in: <em>Differential Characters and Geometric Chains</em>, in: <em>Differential Characters</em>, Lecture Notes in Mathematics <strong>2112</strong>, Springer (2014) [[arXiv:1303.6457](https://arxiv.org/abs/1303.6457), <a href='https://doi.org/10.1007/978-3-319-07034-6_1'>doi:10.1007/978-3-319-07034-6_1</a>]</p> </li> </ul> <p>Transgression of <a class='existingWikiWord' href='/nlab/show/diff/group+cohomology'>group cohomology</a> for <a class='existingWikiWord' href='/nlab/show/diff/discrete+group'>discrete groups</a> to cohomology of <a class='existingWikiWord' href='/nlab/show/diff/inertia+orbifold'>inertia groupoids</a> (see at <em><a class='existingWikiWord' href='/nlab/show/diff/transgression+in+group+cohomology'>transgression in group cohomology</a></em>):</p> <ul> <li id='Willerton08'> <p><a class='existingWikiWord' href='/nlab/show/diff/Simon+Willerton'>Simon Willerton</a>, Section 1 of: <em>The twisted Drinfeld double of a finite group via gerbes and finite groupoids</em>, Algebr. Geom. Topol. 8 (2008) 1419-1457 (<a href='https://arxiv.org/abs/math/0503266'>arXiv:math/0503266</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Jean-Louis+Tu'>Jean-Louis Tu</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ping+Xu'>Ping Xu</a>, Section 3 of: <em>The ring structure for equivariant twisted K-theory</em>, J. Reine Angew. Math. 635 (2009), 97–148 (<a href='https://arxiv.org/abs/math/0604160'>arXiv:math/0604160</a>, <a href='https://doi.org/10.1515/CRELLE.2009.077'>doi:10.1515/CRELLE.2009.077</a>)</p> </li> <li id='AdemRuanZhang07'> <p><a class='existingWikiWord' href='/nlab/show/diff/Alejandro+Adem'>Alejandro Adem</a>, <a class='existingWikiWord' href='/nlab/show/diff/Yongbin+Ruan'>Yongbin Ruan</a>, <a class='existingWikiWord' href='/nlab/show/diff/Bin+Zhang'>Bin Zhang</a>, Section 4 of: <em>A Stringy Product on Twisted Orbifold K-theory</em>, Morfismos (10th Anniversary Issue), <strong>11</strong> 2 (2007) 33-64 [[arXiv:math/0605534](https://arxiv.org/abs/math/0605534), <a href='www.morfismos.cinvestav.mx/Portals/morfismos/SiteDocs/Articulos/Volumen11/No2/Zhang/arz.pdf'>Morfismos pdf</a>]</p> </li> </ul><ins class='diffins'> </ins><ins class='diffins'><p>The <a class='existingWikiWord' href='/nlab/show/diff/image'>image</a> of the transgression operation from <a class='existingWikiWord' href='/nlab/show/diff/bundle+gerbe'>bundle gerbes</a> (<a class='existingWikiWord' href='/nlab/show/diff/connection+on+a+bundle+gerbe'>with connection</a>) to <a class='existingWikiWord' href='/nlab/show/diff/line+bundle'>complex line bundles</a> (<a class='existingWikiWord' href='/nlab/show/diff/connection+on+a+vector+bundle'>with connection</a>) on the <a class='existingWikiWord' href='/nlab/show/diff/free+loop+space'>free loop space</a> of their base space is characterized (as consisting of the “<a class='existingWikiWord' href='/nlab/show/diff/fusion+bundle'>fusion bundles</a>”) in:</p></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Konrad+Waldorf'>Konrad Waldorf</a>, <em>Transgression to Loop Spaces and its Inverse, I: Diffeological Bundles and Fusion Maps</em>, Cah. Topol. Geom. Differ. Categ., 2012, Vol. LIII, 162-210 [[arXiv:0911.3212](https://arxiv.org/abs/0911.3212), <a href='http://cahierstgdc.com/index.php/volumes/volume-liii-2012'>cahierstgdc:LIII</a>]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Konrad+Waldorf'>Konrad Waldorf</a>, <em>Transgression to Loop Spaces and its Inverse, II: Gerbes and Fusion Bundles with Connection</em>, Asian Journal of Mathematics <strong>20</strong> 1 (2016) 59-116 [[arXiv:1004.0031](https://arxiv.org/abs/1004.0031), <a href='https://dx.doi.org/10.4310/AJM.2016.v20.n1.a4'>doi:10.4310/AJM.2016.v20.n1.a4</a>]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Konrad+Waldorf'>Konrad Waldorf</a>, <em>Transgression to Loop Spaces and its Inverse, III: Gerbes and Thin Fusion Bundles</em>, Advances in Mathematics <strong>231</strong> (2012) 3445-3472 [[arXiv:1109.0480](https://arxiv.org/abs/1109.0480), <a href='https://doi.org/10.1016/j.aim.2012.08.016'>doi:10.1016/j.aim.2012.08.016</a>]</p> </li> </ul></ins> <p> </p> <div class='footnotes'><hr /><ol><li id='fn:1'> <p>To the best of <a class='existingWikiWord' href='/nlab/show/diff/Andrew+Stacey'>my</a> knowledge, this remark is attributable to <a class='existingWikiWord' href='/nlab/show/diff/Ralph+Cohen'>Ralph Cohen</a>. <a href='#fnref:1' rev='footnote'>↩</a></p> </li></ol></div> </div> <div class="revisedby"> <p> Last revised on December 11, 2022 at 17:03:52. 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