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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+type+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'>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/%28infinity%2C1%29-category'>(∞,1)-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/homotopy+category+of+an+%28infinity%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/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/homotopy'>right homotopy</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/path+space+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/localization+at+geometric+homotopies'>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+%28infinity%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+%28infinity%2C1%29-topos'>geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-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+infinity-groupoid+in+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid+of+a+locally+infinity-connected+%28infinity%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+%28infinity%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+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> <h4 id='algebraic_topology'>Algebraic topology</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/algebraic+topology'>algebraic topology</a></strong> – application of <a class='existingWikiWord' href='/nlab/show/higher+algebra'>higher algebra</a> and <a class='existingWikiWord' href='/nlab/show/higher+category+theory'>higher category theory</a> to the study of (<a class='existingWikiWord' href='/nlab/show/stable+homotopy+theory'>stable</a>) <a class='existingWikiWord' href='/nlab/show/homotopy+theory'>homotopy theory</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/topological+space'>topological space</a>, <a class='existingWikiWord' href='/nlab/show/homotopy+type'>homotopy type</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/homotopy+group'>homotopy</a></p> <p><a class='existingWikiWord' href='/nlab/show/cohomology'>cohomology</a></p> <p><a class='existingWikiWord' href='/nlab/show/homology'>homology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/spectral+sequence'>spectral sequence</a></p> </li> </ul> </div> </div> </div> <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='#ForTopologicalSpaces'>For topological spaces</a></li><li><a href='#for_simplicial_sets'>For simplicial sets</a></li><li><a href='#for_lie_groupoids'>For Lie groupoids</a></li><li><a href='#for_objects_in_a_general_stack_topos'>For objects in a general <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-stack <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-topos</a></li></ul></li><li><a href='#truncconn'>Truncated and connected spaces</a></li><li><a href='#examples'>Examples</a><ul><li><a href='#in_low_dimensions'>In low dimensions</a></li><li><a href='#of_the_circle'>Of the circle</a></li><li><a href='#of_spheres'>Of spheres</a></li></ul></li><li><a href='#history'>History</a></li><li><a href='#properties'>Properties</a></li><li><a href='#some_general_nonsense'>Some general nonsense</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a><ul><li><a href='#general'>General</a></li><li><a href='#formalization_in_hott'>Formalization in HoTT</a></li></ul></li></ul></div> <h2 id='idea'>Idea</h2> <p>The <em>homotopy groups</em> <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_n(X,x)</annotation></semantics></math> of a <a class='existingWikiWord' href='/nlab/show/pointed+object'>pointed</a> <a class='existingWikiWord' href='/nlab/show/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,x)</annotation></semantics></math> are a sequence of <a class='existingWikiWord' href='/nlab/show/group'>groups</a> that generalise the <a class='existingWikiWord' href='/nlab/show/fundamental+group'>fundamental group</a> <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_1(X,x)</annotation></semantics></math> to higher <a class='existingWikiWord' href='/nlab/show/homotopy'>homotopies</a>.</p> <p>The <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>th homotopy group <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_n(X,x)</annotation></semantics></math> has as elements <a class='existingWikiWord' href='/nlab/show/equivalence+class'>equivalence classes</a> of <a class='existingWikiWord' href='/nlab/show/sphere'>spheres</a> <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi><mo>:</mo><msubsup><mi>S</mi> <mo>*</mo> <mi>n</mi></msubsup><mo>→</mo><msub><mi>X</mi> <mo>*</mo></msub></mrow><annotation encoding='application/x-tex'>\gamma : S_*^n \to X_*</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> where two such are regarded as equivalent if there is a <a class='existingWikiWord' href='/nlab/show/homotopy'>left homotopy</a> <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi><mo>⇒</mo><mi>γ</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>\gamma \Rightarrow \gamma'</annotation></semantics></math> between them, fixing the base point. The group operation is given by gluing of two spheres at their basepoint.</p> <p>In degree 0, <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_0(X,x)</annotation></semantics></math> is not a group but merely a <a class='existingWikiWord' href='/nlab/show/pointed+set'>pointed set</a>. In degree <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>n \geq 2</annotation></semantics></math> all homotopy groups are <a class='existingWikiWord' href='/nlab/show/abelian+group'>abelian groups</a>. Only <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_1(X,x)</annotation></semantics></math> may be an arbitrary group. In general, <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_n(X,x)</annotation></semantics></math> is an <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/k-tuply+groupal+n-groupoid'>tuply groupal</a> set.</p> <p>Lower homotopy groups <a class='existingWikiWord' href='/nlab/show/action'>act</a> on higher homotopy groups; the nonabelian group cohomology of this gives the Postnikov invariants of the space. All of this data put together allows one to reconstruct the original space, at least up to weak <a class='existingWikiWord' href='/nlab/show/homotopy+type'>homotopy type</a>, through its <a class='existingWikiWord' href='/nlab/show/Postnikov+system'>Postnikov system</a>.</p> <p>These definitions only depend on the <a class='existingWikiWord' href='/nlab/show/homotopy+type'>homotopy type</a> of <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, by definition: a <a class='existingWikiWord' href='/nlab/show/weak+homotopy+equivalence'>weak homotopy equivalence</a> between topological spaces is a <a class='existingWikiWord' href='/nlab/show/continuous+map'>continuous map</a> that induces an <a class='existingWikiWord' href='/nlab/show/isomorphism'>isomorphism</a> on all homotopy groups.</p> <p>Accordingly, homotopy groups are defined for all other models of <a class='existingWikiWord' href='/nlab/show/homotopy+type'>homotopy types</a>, notably for <a class='existingWikiWord' href='/nlab/show/simplicial+set'>simplicial sets</a>. See at <em><a class='existingWikiWord' href='/nlab/show/simplicial+homotopy+group'>simplicial homotopy group</a></em> for more.</p> <p>There are generalizations of the concept to <a class='existingWikiWord' href='/nlab/show/homotopy+group+of+a+spectrum'>stable homotopy groups</a> of <a class='existingWikiWord' href='/nlab/show/spectrum'>spectra</a> and to <a class='existingWikiWord' href='/nlab/show/equivariant+homotopy+group'>equivariant homotopy groups</a> for <a class='existingWikiWord' href='/nlab/show/topological+G-space'>topological G-spaces</a> and <a class='existingWikiWord' href='/nlab/show/G-spectrum'>equivariant spectra</a>.</p> <h2 id='Definition'>Definition</h2> <h3 id='ForTopologicalSpaces'>For topological spaces</h3> <p>Let <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/topological+space'>topological space</a>, let <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>:</mo><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x : * \to X</annotation></semantics></math> be a point, to be called the <em>base point</em>.</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}</annotation></semantics></math>, let <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>S^n</annotation></semantics></math> be the pointed <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/sphere'>sphere</a>.</p> <div class='num_defn'> <h6 id='definition_2'>Definition</h6> <p>The underlying set of <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_n(X,x)</annotation></semantics></math> is the set of <a class='existingWikiWord' href='/nlab/show/equivalence+class'>equivalence classes</a> of basepoint-preserving <a class='existingWikiWord' href='/nlab/show/continuous+map'>continuous functions</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi><mo>:</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'> \gamma : S^n \to X </annotation></semantics></math></div> <p>where two such are regarded as equivalent if there is a basepoint-preserving <a class='existingWikiWord' href='/nlab/show/homotopy'>left homotopy</a> between them.</p> </div> <p>Now we will put some structure on that set.</p> <div class='num_defn'> <h6 id='definition_3'>Definition</h6> <p>There are <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> independent equators through the basepoint of <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>S^n</annotation></semantics></math>. Given two maps <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>→</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f, g: S^n \to (X,x)</annotation></semantics></math>, form their <a class='existingWikiWord' href='/nlab/show/coproduct'>copairing</a> in the category of pointed spaces to get a map</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>∨</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>→</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> S^n \vee S^n \to (X,x) </annotation></semantics></math></div> <p>(where <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∨</mo></mrow><annotation encoding='application/x-tex'>\vee</annotation></semantics></math> indicates the <a class='existingWikiWord' href='/nlab/show/wedge+sum'>wedge sum</a>); then combine this with a map <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>∨</mo><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>S^n \to S^n \vee S^n</annotation></semantics></math> that maps the <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math>th equator to the basepoint and each hemisphere to one copy of the sphere. The result is a map <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>→</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>S^n \to (X,x)</annotation></semantics></math>, called the <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math>th <strong>concatenation</strong> of <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><msub><mo>→</mo> <mi>i</mi></msub><msup><mi>S</mi> <mi>n</mi></msup><mo>∨</mo><msup><mi>S</mi> <mi>n</mi></msup><mover><mo>→</mo><mrow><mo stretchy='false'>[</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>]</mo></mrow></mover><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'> S^n \to_i S^n \vee S^n \stackrel{[f,g]}{\to} (X,x) \,. </annotation></semantics></math></div> <p>One can check that each of these operations respects homotopy equivalence and hence equips <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_n(X,x)</annotation></semantics></math> with the structure of a <a class='existingWikiWord' href='/nlab/show/magma'>magma</a>.</p> </div> <div class='num_prop'> <h6 id='proposition'>Proposition</h6> <p>These magmas are in fact <a class='existingWikiWord' href='/nlab/show/group'>groups</a>; in particular:</p> <ul> <li>the <a class='existingWikiWord' href='/nlab/show/constant+function'>constant function</a> that maps all of <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>S^n</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> represents the <strong>null element</strong> of <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_n(X,x)</annotation></semantics></math>, which is an identity for every concatenation.</li> </ul> </div> <div class='num_remark'> <h6 id='remark'>Remark</h6> <p>This is closely related to the statement that the <a class='existingWikiWord' href='/nlab/show/suspensions+are+H-cogroup+objects'>positive dimension spheres are H-cogroup objects</a> (see there) in the <a class='existingWikiWord' href='/nlab/show/homotopy+category'>homotopy category</a> of <a class='existingWikiWord' href='/nlab/show/pointed+topological+space'>pointed topological spaces</a>.</p> </div> <div class='num_defn'> <h6 id='definition_4'>Definition</h6> <p>This may seem like quite a complicated kind of structure, but it is actually quite simple up to homotopy. First of all, all <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> concatenations of given maps <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> are homotopic, so we speak of simply a single <strong>concatenation</strong> for <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n \geq 1</annotation></semantics></math> (and none for <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>n = 0</annotation></semantics></math>). By the <a class='existingWikiWord' href='/nlab/show/Eckmann-Hilton+argument'>Eckmann-Hilton argument</a>, this concatenation will be commutative up to homotopy for <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>n \geq 2</annotation></semantics></math>. In any case, it is <a class='existingWikiWord' href='/nlab/show/associativity'>associative</a> and <a class='existingWikiWord' href='/nlab/show/inverse'>invertible</a> up to homotopy, and the null element is an identity up to homotopy.</p> </div> <div class='num_prop'> <h6 id='proposition_2'>Proposition</h6> <p>The result is that the set <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_n(X,x)</annotation></semantics></math> of equivalence classes is an abelian group for <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>n \geq 2</annotation></semantics></math>, a group for <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n = 1</annotation></semantics></math>, and a pointed set for <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>n = 0</annotation></semantics></math> (when the null element is the only structure).</p> </div> <div class='num_prop'> <h6 id='proposition_3'>Proposition</h6> <p>If <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/connected+space'>path-connected</a>, then all of the <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>a</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_n(X,a)</annotation></semantics></math> are <a class='existingWikiWord' href='/nlab/show/isomorphism'>isomorphic</a>. Accordingly, it's traditional to just write <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_n(X)</annotation></semantics></math> in that case. (This is why we must use <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Π</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Pi_n(X)</annotation></semantics></math> for the homotopy <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-groupoid.) However, there may be many different isomorphisms between <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>a</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_n(X,a)</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_n(X,b)</annotation></semantics></math> (given by <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_{n+1}(X)</annotation></semantics></math>), so a more careful treatment requires keeping track of the basepoint even in the connected case.</p> </div> <h3 id='for_simplicial_sets'>For simplicial sets</h3> <p>See <a class='existingWikiWord' href='/nlab/show/simplicial+homotopy+group'>simplicial homotopy group</a>.</p> <h3 id='for_lie_groupoids'>For Lie groupoids</h3> <ul> <li><a class='existingWikiWord' href='/nlab/show/homotopy+groups+of+a+Lie+groupoid'>homotopy groups of a Lie groupoid</a></li> </ul> <h3 id='for_objects_in_a_general_stack_topos'>For objects in a general <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-stack <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-topos</h3> <p><a class='existingWikiWord' href='/nlab/show/Top'>Top</a> is the archetypical <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-topos'>(∞,1)-topos</a>. The definition of homotopy groups for objects in <a class='existingWikiWord' href='/nlab/show/Top'>Top</a> is just a special case of a general definition of homotopy groups of objects of <a class='existingWikiWord' href='/nlab/show/infinity-stack'>∞-stack</a> <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-topos'>(∞,1)-topos</a>es.</p> <p>This is described in detail at</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/homotopy+groups+in+an+%28infinity%2C1%29-topos'>homotopy groups in an (∞,1)-topos</a>.</li> </ul> <h2 id='truncconn'>Truncated and connected spaces</h2> <p>Often it is useful to talk about spaces whose homotopy groups are all trivial above or below a certain degree, for instance in the context of <a class='existingWikiWord' href='/nlab/show/Postnikov+system'>Postnikov towers</a> and <a class='existingWikiWord' href='/nlab/show/Whitehead+tower'>Whitehead towers</a>.</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>=</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>…</mi><mo>,</mo><mn>∞</mn></mrow><annotation encoding='application/x-tex'>n = -1, 0, 1, 2, \ldots, \infty</annotation></semantics></math>:</p> <ul> <li> <p>a space is <strong><math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/homotopy+n-type'>truncated</a></strong> if all homotopy groups <em>above</em> degree <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> are trivial.</p> </li> <li> <p>a space is <strong><math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/n-connected+space'>connected</a></strong> (or <strong><math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-simply connected</strong>) if it is <a class='existingWikiWord' href='/nlab/show/inhabited+set'>inhabited</a> and all homotopy groups <em>at or below</em> degree <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> are trivial.</p> </li> </ul> <p>Vacuously, every space is <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-truncated, and precisely the inhabited spaces are <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(-1)</annotation></semantics></math>-connected. On the other end, precisely the weakly <a class='existingWikiWord' href='/nlab/show/contractible+space'>contractible spaces</a> are <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-connected, and a space is <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(-1)</annotation></semantics></math>-truncated iff it is weakly contractible if inhabited. (So <a class='existingWikiWord' href='/nlab/show/classical+mathematics'>classically</a>, using <a class='existingWikiWord' href='/nlab/show/excluded+middle'>excluded middle</a>, a space is <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(-1)</annotation></semantics></math>-truncated iff it is either <a class='existingWikiWord' href='/nlab/show/empty+space'>empty</a> or weakly contractible.)</p> <p>To extend one step further in <a class='existingWikiWord' href='/nlab/show/negative+thinking'>negative thinking</a>, every space (even the empty space) is <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(-2)</annotation></semantics></math>-connected, and precisely a weakly contractible space (but not the empty space) is <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(-2)</annotation></semantics></math>-truncated.</p> <ul> <li>A <a class='existingWikiWord' href='/nlab/show/pointed+space'>pointed space</a> is a <strong>degree-<math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/Eilenberg-Mac+Lane+space'>Eilenberg-Mac Lane space</a></strong> if it is both <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-truncated as well as <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(n-1)</annotation></semantics></math>-connected, i.e. if its only nontrivial homotopy group is in degree <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>.</li> </ul> <p>A weakly contractible space is an Eilenberg–MacLane space in every degree, and these are the only Eilenberg–MacLane spaces in degree <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>-1</annotation></semantics></math>. In degree <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math>, they are the pointed <a class='existingWikiWord' href='/nlab/show/discrete+object'> discrete spaces</a> (and those weakly homotopy equivalent to such). In degree <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>, they are (up to weak homotopy equivalence) precisely the <a class='existingWikiWord' href='/nlab/show/classifying+space'>classifying spaces</a> of <a class='existingWikiWord' href='/nlab/show/group'>groups</a>. And so on.</p> <h2 id='examples'>Examples</h2> <h3 id='in_low_dimensions'>In low dimensions</h3> <p>The <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math>th homotopy ‘group’ <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>a</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_0(X,a)</annotation></semantics></math> can be identified with the set of all <a class='existingWikiWord' href='/nlab/show/connected+space'> path components</a> of <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, with the component containing <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> as the basepoint. Similarly, the <a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid'>fundamental 0-groupoid</a> <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Pi_0(X)</annotation></semantics></math> is the set of all path components without a chosen basepoint. Note that <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Pi_0(X)</annotation></semantics></math> is traditionally written <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_0(X)</annotation></semantics></math>, even without a basepoint.</p> <p>The <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>st homotopy group <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>a</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_1(X,a)</annotation></semantics></math> is precisely the <a class='existingWikiWord' href='/nlab/show/fundamental+group'>fundamental group</a> of <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> at <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math>. This is the original example from which all others derived. It was once written simply <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>π</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>a</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi(X,a)</annotation></semantics></math> with the <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>π</mi></mrow><annotation encoding='application/x-tex'>\pi</annotation></semantics></math> standing for Poincaré, who invented it.</p> <div class='query'> <p>At least, that's where I <em>think</em> that it comes from … —Toby</p> </div> <h3 id='of_the_circle'>Of the circle</h3> <p>The first homotopy group of the <a class='existingWikiWord' href='/nlab/show/circle'>circle</a> <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_94' 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> is the group of <a class='existingWikiWord' href='/nlab/show/integer+'> integers</a>.</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy='false'>)</mo><mo>≃</mo><mi>ℤ</mi><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'> \pi_1(S^1) \simeq \mathbb{Z} \,. </annotation></semantics></math></div> <p>A formalization of a proof of this in <a class='existingWikiWord' href='/nlab/show/homotopy+type+theory'>homotopy type theory</a> is in (<a href='#ShulmanPi1S1'>Shulman</a>).</p> <h3 id='of_spheres'>Of spheres</h3> <p>See <a class='existingWikiWord' href='/nlab/show/homotopy+groups+of+spheres'>homotopy groups of spheres</a>.</p> <h2 id='history'>History</h2> <p>In the early years of the 20th century it was known that the nonabelian fundamental group <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>a</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_1(X,a)</annotation></semantics></math> of a space <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> with base point <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> was useful in geometry and complex analysis. It was also known that the abelian <a class='existingWikiWord' href='/nlab/show/homology'> homology groups</a> <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H_n(X)</annotation></semantics></math> existed for all <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>n \geq 0</annotation></semantics></math> and that if <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is connected then <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H_1(X)</annotation></semantics></math> is isomorphic to the abelianisation of any <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>a</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_1(X,a)</annotation></semantics></math>.</p> <p>Consequently it was hoped to generalise the fundamental group to higher dimensions, producing nonabelian groups whose abelianisations would be the homology groups.</p> <p>In 1932, E. Čech proposed a definition of higher homotopy groups using maps of spheres, but the paper was rejected for the Zurich ICM since it was found that these groups <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>a</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_n(X,a)</annotation></semantics></math> were abelian for <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>n \geq 2</annotation></semantics></math>, and so do not generalise the fundamental group in the way that was originally desired. Nonetheless, they have proved to be extremely important in homotopy theory, although more difficult to compute in general than homology groups. See <a class='existingWikiWord' href='/nlab/show/weak+homotopy+equivalence'>weak homotopy equivalence</a>.</p> <h2 id='properties'>Properties</h2> <p>It was early realised that the fundamental groupoid <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Pi_1(X)</annotation></semantics></math> operates on the family of groups <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>a</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo><mi>a</mi><mo>∈</mo><mi>X</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{\pi_n(X,a) | a \in X\}</annotation></semantics></math> which should thus together be regarded as a module over <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>a</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_1(X,a)</annotation></semantics></math>.</p> <p>A key property of homotopy groups is the <em>Whitehead theorem</em>: if <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f:X \to Y</annotation></semantics></math> is a map of connected <a class='existingWikiWord' href='/nlab/show/m-cofibrant+space'>m-cofibrant spaces</a> (spaces each of the homotopy type of a <a class='existingWikiWord' href='/nlab/show/CW+complex'>CW complex</a>), and <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> induces isomorphisms <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>a</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_n(X,a) \to \pi_n(Y,f(a))</annotation></semantics></math> for some <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> and all <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n \geq 1</annotation></semantics></math>, then <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/homotopy+equivalence'>homotopy equivalence</a>.</p> <p>However, the homotopy groups by themselves, even considering the operations of <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>\pi_1</annotation></semantics></math>, do not characterise homotopy types. See also <a class='existingWikiWord' href='/nlab/show/algebraic+homotopy'>algebraic homotopy theory</a>.</p> <p>See also the <em><a class='existingWikiWord' href='/nlab/show/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></em>.</p> <h2 id='some_general_nonsense'>Some general nonsense</h2> <p>Using the <a class='existingWikiWord' href='/nlab/show/Eckmann-Hilton+duality'>Eckmann-Hilton duality</a> between <a class='existingWikiWord' href='/nlab/show/cohomology'>cohomology</a> and <a class='existingWikiWord' href='/nlab/show/homotopy+%28as+an+operation%29'>homotopy (as an operation)</a> one may discuss homotopy groups along the same lines as the discussion of <a class='existingWikiWord' href='/nlab/show/cohomology+group'>cohomology groups</a> (see there).</p> <p>From that perspective we might say that:</p> <p>for <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo>,</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>B, X</annotation></semantics></math> any two objects in an <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-topos'>(∞,1)-topos</a> <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math>, the “homotopy of <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> with co-coefficients <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>” is the <a class='existingWikiWord' href='/nlab/show/hom-set'>hom-set</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mi>H</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo><mo>:</mo><mo>=</mo><mi>H</mi><mo stretchy='false'>(</mo><mi>B</mi><mo>,</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>:</mo><mo>=</mo><msub><mi>π</mi> <mn>0</mn></msub><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mi>B</mi><mo>,</mo><mi>X</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace'></mspace><mo>,</mo></mrow><annotation encoding='application/x-tex'> \pi_H(X,B) := H(B,X) := \pi_0 \mathbf{H}(B,X) \,, </annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mi>H</mi></msub></mrow><annotation encoding='application/x-tex'>\pi_H</annotation></semantics></math> denotes the <a class='existingWikiWord' href='/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category'>homotopy category</a> of <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math>.</p> <p>For the special case that the object <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> here is a <a class='existingWikiWord' href='/nlab/show/cogroup'>co-group object</a>, this <em>homotopy set</em> <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>π</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi(X,B)</annotation></semantics></math> naturally inherits the structure of a <a class='existingWikiWord' href='/nlab/show/group'>group</a>.</p> <p>The standard example is that where <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo>=</mo><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>B = S^n</annotation></semantics></math> is the <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/sphere'>sphere</a>. This naturally comes with an co-group structure up to homotopy, which is precisely the structure underlying the co-category structure of the <a class='existingWikiWord' href='/nlab/show/interval+object'>interval object</a> and more generally that underlying the mechanism of the <a class='existingWikiWord' href='/nlab/show/Trimble+n-category'>Trimble n-category</a>.</p> <p>As opposed to <a class='existingWikiWord' href='/nlab/show/cohomology'>cohomology</a> where people are used to talking about <a class='existingWikiWord' href='/nlab/show/generalized+cohomology'>generalized cohomology</a>, “homotopy” usually just means this ordinary homotopy for <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo>=</mo><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>B = S^n</annotation></semantics></math>.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/homotopy+group+of+a+spectrum'>homotopy group of a spectrum</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/weak+homotopy+equivalence'>weak homotopy equivalence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/localization+of+a+space'>p-localization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Goodwillie+calculus'>Goodwillie calculus</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/homotopy+groups+in+an+%28infinity%2C1%29-topos'>homotopy groups in an (infinity,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/finite+homotopy+type'>finite homotopy type</a>, <a class='existingWikiWord' href='/nlab/show/nilpotent+topological+space'>nilpotent homotopy type</a>, <a class='existingWikiWord' href='/nlab/show/localization+of+a+space'>p-local homotopy type</a>, <a class='existingWikiWord' href='/nlab/show/p-adic+completion'>p-complete homotopy type</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/equivariant+homotopy+group'>equivariant homotopy group</a></p> </li> </ul> <h2 id='references'>References</h2> <h3 id='general'>General</h3> <p>The oiginal definition:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/Witold+Hurewicz'>Witold Hurewicz</a>: <em>Beiträge zur Topologie der Deformationen (I. Höherdimensionale Homotopiegruppen)</em>, Proc. Akad. Wet. Amsterdam <strong>38</strong> (1935) 112–119 [[pdf](https://dwc.knaw.nl/DL/publications/PU00016667.pdf), <a class='existingWikiWord' href='/nlab/files/Hurewicz-BeitraegeI.pdf' title='pdf'>pdf</a>]</li> </ul> <p>On the early history of the notion of the higher homotopy groups (beyond the <a class='existingWikiWord' href='/nlab/show/fundamental+group'>fundamental group</a>):</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/Peter+Hilton'>Peter Hilton</a>, p. 289 (9 of 11) in: <em>Subjective History of Homology and Homotopy Theory</em>, Mathematics Magazine <strong>61</strong> 5 (1988) 282-291 [[doi:10.2307/2689545](https://doi.org/10.2307/2689545)]</li> </ul> <blockquote> <p>“At a meeting in Vienna in 1931 <a class='existingWikiWord' href='/nlab/show/Eduard+%C4%8Cech'>Čech</a> gave a paper in which he described certain groups from the homotopy point of view. He had no applications of these groups. Moreover, he had only one theorem, that they were <a class='existingWikiWord' href='/nlab/show/abelian+group'>commutative</a>. And he was persuaded by people, and we know that <a class='existingWikiWord' href='/nlab/show/Pavel+Aleksandrov'>Alexandroff</a> played a role here, that they could not be interesting, because it was thought that any information that could be obtained from <a class='existingWikiWord' href='/nlab/show/abelian+group'>Abelian groups</a> must come from the <a class='existingWikiWord' href='/nlab/show/ordinary+homology'>homology</a>. <a class='existingWikiWord' href='/nlab/show/Witold+Hurewicz'>Hurewicz</a> redefined the <a class='existingWikiWord' href='/nlab/show/homotopy+group'>homotopy groups</a> and immediately gave important applications in a series of four notes which were intended as preliminary publications. In that series of four papers he showed the significance of what we now call <a class='existingWikiWord' href='/nlab/show/obstruction'>obstruction theory</a>.”</p> </blockquote> <p>Modern accounts (for more see at <em><a class='existingWikiWord' href='/nlab/show/homotopy+theory'>homotopy theory</a></em>):</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/Paul+Goerss'>Paul Goerss</a>, <a class='existingWikiWord' href='/nlab/show/John+Frederick+Jardine'>J. F. Jardine</a>, section I.7 of: <em><a class='existingWikiWord' href='/nlab/show/Simplicial+homotopy+theory'>Simplicial homotopy theory</a></em>, Progress in Mathematics, Birkhäuser (1996)</p> </li> <li id='AGP02'> <p>Marcelo Aguilar, <a class='existingWikiWord' href='/nlab/show/Samuel+Gitler'>Samuel Gitler</a>, Carlos Prieto, section 3.4 of <em>Algebraic topology from a homotopical viewpoint</em>, Springer (2002) (<a href='http://tocs.ulb.tu-darmstadt.de/106999419.pdf'>toc pdf</a>)</p> </li> <li id='Hutchings'> <p><a class='existingWikiWord' href='/nlab/show/Michael+Hutchings'>Michael Hutchings</a>, <em>Introduction to higher homotopy groups and obstruction theory</em> (<a href='http://math.berkeley.edu/~hutching/teach/215b-2011/homotopy.pdf'>pdf</a>)</p> </li> </ul> <p>With an eye towards application in <a class='existingWikiWord' href='/nlab/show/mathematical+physics'>mathematical physics</a>:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/Mikio+Nakahara'>Mikio Nakahara</a>, Chapter 4 of: <em><a class='existingWikiWord' href='/nlab/show/Geometry%2C+Topology+and+Physics'>Geometry, Topology and Physics</a></em>, IOP (2003) [[doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), <a href='http://alpha.sinp.msu.ru/~panov/LibBooks/GRAV/(Graduate_Student_Series_in_Physics)Mikio_Nakahara-Geometry,_Topology_and_Physics,_Second_Edition_(Graduate_Student_Series_in_Physics)-Institute_of_Physics_Publishing(2003).pdf'>pdf</a>]</li> </ul> <h3 id='formalization_in_hott'>Formalization in HoTT</h3> <p>Homotopy groups and their properties can naturally be formalized in <a class='existingWikiWord' href='/nlab/show/homotopy+type+theory'>homotopy type theory</a>. In this context a proof that <math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy='false'>)</mo><mo>≃</mo><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>\pi_1(S^1)\simeq \mathbb{Z}</annotation></semantics></math> is in</p> <ul> <li id='ShulmanPi1S1'> <p><a class='existingWikiWord' href='/nlab/show/Mike+Shulman'>Mike Shulman</a>, <em><a href='https://github.com/HoTT/HoTT/blob/master/Coq/HIT/Pi1S1.v'>P1S1.v</a></em></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Daniel+Licata'>Dan Licata</a>, <a class='existingWikiWord' href='/nlab/show/Mike+Shulman'>Mike Shulman</a>, <em>Calculating the fundamental group of the circle in homotopy type theory</em> (<a href='http://www.cs.cmu.edu/~drl/pubs/ls13pi1s1/ls13pi1s1.pdf'>pdf</a>)</p> </li> </ul> <p>and a proof that</p> <p><math class='maruku-mathml' display='inline' id='mathml_eed42f0c878acbcb1df1666d6fa7e923ea93a6c8_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy='false'>(</mo><msup><mi>S</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo><mo>≃</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mi>for</mi><mspace width='thickmathspace'></mspace><mi>k</mi><mo><</mo><mi>n</mi></mtd></mtr> <mtr><mtd><mi>ℤ</mi></mtd> <mtd><mi>for</mi><mspace width='thickmathspace'></mspace><mi>k</mi><mo>=</mo><mi>n</mi></mtd></mtr></mtable></mrow></mrow></mrow><annotation encoding='application/x-tex'>\pi_k(S^n) \simeq \left\{ \array{ 0 & for \;k \lt n \\ \mathbb{Z} & for\; k = n} \right.</annotation></semantics></math></p> <p>is in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/Daniel+Licata'>Dan Licata</a>, <a class='existingWikiWord' href='/nlab/show/Guillaume+Brunerie'>Guillaume Brunerie</a>, <em><a href='https://github.com/dlicata335/hott-agda/blob/master/homotopy/PiNSN.agda'>PiNSN.agda</a></em></li> </ul> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> </div>  </div>  </body> </html>