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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" 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"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#the_category__of_homotopy_operations'>The category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> of homotopy operations</a></li> <ul> <li><a href='#properties'>Properties</a></li> </ul> <li><a href='#algebras'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebras</a></li> <ul> <li><a href='#properties_2'>Properties</a></li> </ul> <li><a href='#the_homotopy_algebra_of_a_pointed_topological_space'>The homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebra of a pointed topological space.</a></li> <li><a href='#the_realisability_problem'>The realisability problem</a></li> <ul> <li><a href='#theorem_blanc_1995'>Theorem (Blanc 1995)</a></li> <li><a href='#example_blanc_1995'>Example (Blanc 1995)</a></li> </ul> <li><a href='#simply_connected_algebras'>Simply connected <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebras</a></li> <li><a href='#truncated_algebras'>Truncated <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebras</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebra</em> is an algebraic model for the <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>*</mo></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">\pi_*X</annotation></semantics></math> of a pointed <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, together with the <a class="existingWikiWord" href="/nlab/show/action">action</a> of the <a class="existingWikiWord" href="/nlab/show/primary+homotopy+operation">primary homotopy operation</a>s on them, in the same sense that algebras over the <a class="existingWikiWord" href="/nlab/show/Steenrod+algebra">Steenrod algebra</a> are models for the <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of a space.</p> <p>Constructions of this type exist in many pointed model categories. It suffices to have a collection of <a class="existingWikiWord" href="/nlab/show/spherical+object">spherical object</a>s.</p> <h2 id="the_category__of_homotopy_operations">The category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> of homotopy operations</h2> <p>The <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> of homotopy operations has</p> <ul> <li> <p>as <a class="existingWikiWord" href="/nlab/show/object">object</a>s - pointed <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a>es with the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of a finite <a class="existingWikiWord" href="/nlab/show/wedge+product">wedge product</a> of <a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>s of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\geq 1</annotation></semantics></math>;</p> </li> <li> <p>as <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>s - homotopy classes of (pointed) <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>s between them.</p> </li> </ul> <h3 id="properties">Properties</h3> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> is a pointed category and has finite coproducts (given by the finite wedges), but not products.</p> </li> <li> <p>There is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a>, <em>smash product</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>Π</mi><mo>×</mo><mi>Π</mi><mo>→</mo><mi>Π</mi></mrow><annotation encoding="application/x-tex">i : \Pi\times \Pi \to \Pi</annotation></semantics></math>, which sends an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U,V)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∧</mo><mi>V</mi><mo>=</mo><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><mo>*</mo><mo stretchy="false">)</mo><mo>∨</mo><mo stretchy="false">(</mo><mo>*</mo><mo>×</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U\wedge V = (U\times V)/((U\times *)\vee(*\times V))</annotation></semantics></math>, which preserves <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>s in each variable.</p> </li> </ul> <p>This category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Π</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\Pi^{op}</annotation></semantics></math> is a finite product theory, in the sense of <a class="existingWikiWord" href="/nlab/show/algebraic+theories">algebraic theories</a> whose models are:</p> <h2 id="algebras"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebras</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Set</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">Set_*</annotation></semantics></math> denote the category of <a class="existingWikiWord" href="/nlab/show/pointed+sets">pointed sets</a>.</p> <div> <h6 id="definition">Definition</h6> <p>A <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebra</em> is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>:</mo><msup><mi>Π</mi> <mi>op</mi></msup><mo>→</mo><msub><mi>Set</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">A: \Pi^{op}\to Set_*</annotation></semantics></math>, which sends <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>s to <a class="existingWikiWord" href="/nlab/show/products">products</a>.</p> <p>A morphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebras is a natural transformation between the corresponding functors.</p> </div> <h3 id="properties_2">Properties</h3> <ul> <li> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>*</mo><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">A* = *</annotation></semantics></math>.</p> </li> <li> <p>The values of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> are determined by the values <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>n</mi></msub><mo>=</mo><mi>A</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A_n = A(S^n)</annotation></semantics></math>, that it takes on the spheres, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^n</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n\geq 1</annotation></semantics></math>.</p> </li> <li> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebra can be considered to be a graded <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>A</mi> <mi>n</mi></msub><msubsup><mo stretchy="false">}</mo> <mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow> <mn>∞</mn></msubsup></mrow><annotation encoding="application/x-tex">\{A_n\}_{n=1}^\infty</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">A_n</annotation></semantics></math> abelian for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>></mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n\gt 1</annotation></semantics></math>, together with</p> <ul> <li>‘<a class="existingWikiWord" href="/nlab/show/Whitehead+product">Whitehead product</a>’ homomorphisms :</li> </ul> </li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>:</mo><msub><mi>A</mi> <mi>p</mi></msub><mo>⊗</mo><msub><mi>A</mi> <mi>q</mi></msub><mo>→</mo><msub><mi>A</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">[-,-] : A_p\otimes A_q \to A_{p+q-1}</annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">p,q \geq 1</annotation></semantics></math> (the case where they are equal to 1 needs special mention, see below.)</p> <ul> <li>‘<a class="existingWikiWord" href="/nlab/show/composition+operation">composition operation</a>s’, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>⋅</mo><mi>α</mi><mo>:</mo><msub><mi>A</mi> <mi>p</mi></msub><mo>→</mo><msub><mi>A</mi> <mi>r</mi></msub></mrow><annotation encoding="application/x-tex">-\cdot \alpha : A_p\to A_r</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>∈</mo><msub><mi>π</mi> <mi>r</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>p</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha \in \pi_r(S^p)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mi>r</mi></mrow><annotation encoding="application/x-tex">1\lt p\lt r</annotation></semantics></math>,</li> </ul> <p>which satisfy the identities that hold for the Whitehead products and composition operations on the higher homotopy groups of a pointed space, and</p> <ul> <li>a left action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A_1</annotation></semantics></math> on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">A_n</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>></mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n\gt 1</annotation></semantics></math>, which commutes with these operations.</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/Whitehead+products">Whitehead products</a> include</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>ξ</mi><mo>,</mo><mi>a</mi><mo stretchy="false">]</mo><mo>=</mo><msup><mrow></mrow> <mi>ξ</mi></msup><mi>a</mi><mo>−</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">[\xi,a] = {}^\xi a - a</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mrow></mrow> <mi>ξ</mi></msup><mi>a</mi></mrow><annotation encoding="application/x-tex">{}^\xi a</annotation></semantics></math> is the result of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A_1</annotation></semantics></math>-action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ξ</mi><mo>∈</mo><msub><mi>A</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\xi \in A_1</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><msub><mi>A</mi> <mi>r</mi></msub></mrow><annotation encoding="application/x-tex">a\in A_r</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>></mo><mn>1</mn></mrow><annotation encoding="application/x-tex">r\gt 1</annotation></semantics></math>; similarly for a right action;</p> </li> <li> <p>the commutators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">]</mo><mo>=</mo><mi>a</mi><mi>b</mi><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msup><mi>b</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">[a,b] = a b a^{-1} b^{-1}</annotation></semantics></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><msub><mi>A</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">a,b \in A_1</annotation></semantics></math>.</p> </li> </ul> <h2 id="the_homotopy_algebra_of_a_pointed_topological_space">The homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebra of a pointed topological space.</h2> <p>For a pointed space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>Π</mi></mrow><annotation encoding="application/x-tex">U \in \Pi</annotation></semantics></math>, define a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>*</mo></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">\pi_* X</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>*</mo></msub><mi>X</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><mi>U</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">\pi_* X(U) = [U,X]_*</annotation></semantics></math>, the set of pointed homotopy classes of pointed maps from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>This is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebra called the <strong>homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebra</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <h2 id="the_realisability_problem">The realisability problem</h2> <p>Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>:</mo><mi>Π</mi><mo>→</mo><msub><mi>sets</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">A: \Pi \to sets_*</annotation></semantics></math> is an abstract <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebra, the <strong>realisability problem</strong> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is to construct, if possible, a pointed space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>≃</mo><msub><mi>π</mi> <mo>*</mo></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">A\simeq \pi_* X</annotation></semantics></math>. The space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is called a <em>realisation</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <p>Things can be complicated!</p> <ol> <li> <p>The homotopy type of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is not always determined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> (hence ‘a’ rather than ‘the“ realisation) , so that raises the additional problem of classifying the realisations.</p> </li> <li> <p>Not all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebras can be realised, in fact</p> </li> </ol> <h4 id="theorem_blanc_1995">Theorem (Blanc 1995)</h4> <p>Given a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebra, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, there is a sequence of <span class="newWikiWord">higher homotopy operation<a href="/nlab/new/higher+homotopy+operation">?</a></span>s depending only on maps between wedges of spheres, such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is realisable if and only if the operations vanish coherently.</p> <h4 id="example_blanc_1995">Example (Blanc 1995)</h4> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>≠</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">p\neq 2</annotation></semantics></math>, a prime and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>≥</mo><mn>4</mn><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r\geq 4(p-1)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>*</mo></msub><msup><mi>S</mi> <mi>r</mi></msup><mo>⊗</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">\pi_*S^r \otimes \mathbb{Z}/p</annotation></semantics></math> cannot be realised (and if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">p = 2</annotation></semantics></math>, one uses <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>≥</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">r\geq 6</annotation></semantics></math>).</p> <p>(There is a problem, discussed in Blanc’s 1995 paper, that the composition operations need not be homomorphisms, so tensoring with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}/p</annotation></semantics></math> has to be interpreted carefully.)</p> <h2 id="simply_connected_algebras">Simply connected <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebras</h2> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebra, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, is said to be <strong>simply connected</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">A_1= 0</annotation></semantics></math>.</p> <p>In this case the universal identities for the <a class="existingWikiWord" href="/nlab/show/primary+homotopy+operations">primary homotopy operations</a> can be described more easily (see Blanc 1993). These include the structural information that the <a class="existingWikiWord" href="/nlab/show/Whitehead+products">Whitehead products</a> make <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> into a graded <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie ring</a> (with a shift of indices).</p> <h2 id="truncated_algebras">Truncated <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebras</h2> <p>The beginnings of a classification theory for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-algebras can be found in Frankland’s thesis (link given below).</p> <h2 id="references">References</h2> <ul> <li>C.R. Stover, <em>A Van Kampen spectral sequence for higher homotopy groups</em>, Topology 29 (1990) 9 - 26.</li> </ul> <p><a class="existingWikiWord" href="/nlab/show/David+Blanc">David Blanc</a> has written a lot on the theory of these objects. An example is</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/David+Blanc">David Blanc</a>, <em>Loop spaces and homotopy operations</em>, Fund. Math. 154 (1997) 75 - 95.</li> </ul> <p>The realisability problem is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/David+Blanc">David Blanc</a>, <em>Higher homotopy operations and the realizability of homotopy groups</em>, Proc. London Math. Soc. (3) 70 (1995) 214 -240,</li> </ul> <p>and further in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/David+Blanc">David Blanc</a>, <em>Algebraic invariants for homotopy types</em>, Math. Proc. Camb. Phil. Soc. 127 (3)(1999) 497 - 523. (preprint version on the <a href="http://arxiv.org/abs/math/9812035">ArXiv</a>.)</li> </ul> <p>There are more recent results on the realisability problem in <a class="existingWikiWord" href="/nlab/show/Martin+Frankland">Martin Frankland</a>‘s <a href="http://www.math.uiuc.edu/~franklan/Frankland_Thesis_20100513.pdf">thesis</a>.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on June 11, 2022 at 10:42:00. 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