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12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> fundamental group (changes) </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/diff/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/4086/#Item_8" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #33 to #34: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='homotopy_theory'>Homotopy theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a></strong></p> <p>flavors: <a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+theory'>stable</a>, <a class='existingWikiWord' href='/nlab/show/diff/equivariant+homotopy+theory'>equivariant</a>, <a class='existingWikiWord' href='/nlab/show/diff/rational+homotopy+theory'>rational</a>, <a class='existingWikiWord' href='/nlab/show/diff/p-adic+homotopy+theory'>p-adic</a>, <a class='existingWikiWord' href='/nlab/show/diff/proper+homotopy+theory'>proper</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometric+homotopy+type+theory'>geometric</a>, <a class='existingWikiWord' href='/nlab/show/diff/cohesive+homotopy+theory'>cohesive</a>, <a class='existingWikiWord' href='/nlab/show/diff/directed+homotopy+theory'>directed</a>…</p> <p>models: <a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological</a>, <a class='existingWikiWord' href='/nlab/show/diff/simplicial+homotopy+theory'>simplicial</a>, <a class='existingWikiWord' href='/nlab/show/diff/localic+homotopy+theory'>localic</a>, …</p> <p>see also <strong><a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology+--+2'>Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Homotopy+Theory'>Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/homotopy'>homotopy</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+homotopy'>higher homotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy type</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Pi-algebra'>Pi-algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/spherical+object'>spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coherent+category+theory'>homotopy coherent category theory</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopical+category'>homotopical category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category+of+fibrant+objects'>category of fibrant objects</a>, <a class='existingWikiWord' href='/nlab/show/diff/cofibration+category'>cofibration category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Waldhausen+category'>Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Ho%28Top%29'>Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/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/diff/homotopy'>left homotopy</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>right homotopy</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/path+space+object'>path object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/generalized+universal+bundle'>universal bundle</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/interval+object'>interval object</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/localization+at+geometric+homotopies'>homotopy localization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/homotopy+group'>homotopy group</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group+of+a+topos'>fundamental group of a topos</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Brown-Grossman+homotopy+group'>Brown-Grossman homotopy group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/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/diff/fundamental+infinity-groupoid'>fundamental ∞-groupoid</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+groupoid'>fundamental groupoid</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/path+groupoid'>path groupoid</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/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/diff/fundamental+%28infinity%2C1%29-category'>fundamental (∞,1)-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fundamental+category'>fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/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/diff/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Blakers-Massey+theorem'>Blakers-Massey theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/higher+homotopy+van+Kampen+theorem'>higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nerve+theorem'>nerve theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+theorem'>Hurewicz theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Galois+theory'>Galois theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+hypothesis'>homotopy hypothesis</a>-theorem</p> </li> </ul> </div> <h4 id='group_theory'>Group Theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/group+theory'>group theory</a></strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/group'>group</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-group'>∞-group</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/group+object'>group object</a>, <a class='existingWikiWord' href='/nlab/show/diff/groupoid+object+in+an+%28infinity%2C1%29-category'>group object in an (∞,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/abelian+group'>abelian group</a>, <a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/super+abelian+group'>super abelian group</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/action'>group action</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-action'>∞-action</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/representation'>representation</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-representation'>∞-representation</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/progroup'>progroup</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/homogeneous+space'>homogeneous space</a></li> </ul> <p><strong>Classical groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/general+linear+group'>general linear group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/unitary+group'>unitary group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/special+unitary+group'>special unitary group</a>. <a class='existingWikiWord' href='/nlab/show/diff/projective+unitary+group'>projective unitary group</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/orthogonal+group'>orthogonal group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/special+orthogonal+group'>special orthogonal group</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symplectic+group'>symplectic group</a></p> </li> </ul> <p><strong>Finite groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/finite+group'>finite group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symmetric+group'>symmetric group</a>, <a class='existingWikiWord' href='/nlab/show/diff/cyclic+group'>cyclic group</a>, <a class='existingWikiWord' href='/nlab/show/diff/braid+group'>braid group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classification+of+finite+simple+groups'>classification of finite simple groups</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sporadic+finite+simple+group'>sporadic finite simple groups</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Monster+group'>Monster group</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mathieu+group'>Mathieu group</a></li> </ul> </li> </ul> <p><strong>Group schemes</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebraic+group'>algebraic group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/abelian+variety'>abelian variety</a></p> </li> </ul> <p><strong>Topological groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+topological+group'>compact topological group</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+group'>locally compact topological group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/maximal+compact+subgroup'>maximal compact subgroup</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/string+group'>string group</a></p> </li> </ul> <p><strong>Lie groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie+group'>Lie group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+Lie+group'>compact Lie group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kac-Moody+group'>Kac-Moody group</a></p> </li> </ul> <p><strong>Super-Lie groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/supergroup'>super Lie group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/super+Euclidean+group'>super Euclidean group</a></p> </li> </ul> <p><strong>Higher groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-group'>2-group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/crossed+module'>crossed module</a>, <a class='existingWikiWord' href='/nlab/show/diff/strict+2-group'>strict 2-group</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/n-group'>n-group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-group'>∞-group</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simplicial+group'>simplicial group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/crossed+complex'>crossed complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/k-tuply+groupal+n-groupoid'>k-tuply groupal n-groupoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/circle+n-group'>circle n-group</a>, <a class='existingWikiWord' href='/nlab/show/diff/string+2-group'>string 2-group</a>, <a class='existingWikiWord' href='/nlab/show/diff/fivebrane+6-group'>fivebrane Lie 6-group</a></p> </li> </ul> <p><strong>Cohomology and Extensions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/group+cohomology'>group cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/group+extension'>group extension</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-group+extension'>∞-group extension</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ext'>Ext-group</a></p> </li> </ul> <p><strong>Related concepts</strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/quantum+group'>quantum group</a></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></li><li><a href='#properties'>Properties</a><ul><li><a href='#naturality'>Naturality</a></li><li><a href='#RelationToSingularHomology'>Relation to singular homology</a></li><li><a href='#relation_to_universal_covers_and_galois_groups'>Relation to universal covers and Galois groups</a></li></ul></li><li><a href='#generalizations'>Generalizations</a><ul><li><a href='#nonlocally_nice_spaces_and_generalised_spaces'>Non-locally ‘nice’ spaces and ‘generalised’ spaces</a></li><li><a href='#proper_fundamental_groups'>Proper fundamental groups</a></li></ul></li><li><a href='#Examples'>Examples</a></li><li><a href='#related_concept'>Related concept</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>The <em>fundamental group</em> <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_1' 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> of a <a class='existingWikiWord' href='/nlab/show/diff/pointed+object'>pointed</a> <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_2' 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> is the group of based <a class='existingWikiWord' href='/nlab/show/diff/homotopy+class'>homotopy classes</a> of <a class='existingWikiWord' href='/nlab/show/diff/loop'>loops</a> at <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>, with multiplication defined by concatenation (following one path by another).</p> <p>This is also called <em>the first <a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a></em> of <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> <p>The notion of <em>fundamental group</em> <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_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> generalises in one direction to the <a class='existingWikiWord' href='/nlab/show/diff/fundamental+groupoid'>fundamental groupoid</a> <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_6' 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>, or in another direction to the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy groups</a> <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_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>a</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_n(X,a)</annotation></semantics></math> for <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_8' 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>. Both of this is contained within the <a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid'>fundamental ∞-groupoid</a> <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Π</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Pi(X)</annotation></semantics></math>.</p> <h2 id='definition'>Definition</h2> <div class='num_defn'> <h6 id='definition_2'>Definition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> and <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_11' 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> a <a class='existingWikiWord' href='/nlab/show/diff/point'>point</a>. A <a class='existingWikiWord' href='/nlab/show/diff/loop+space'>loop</a> in <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> based at <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_13' 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 <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi><mo>:</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'> \gamma : \Delta^1 \to X </annotation></semantics></math></div> <p>from the topological 1-<a class='existingWikiWord' href='/nlab/show/diff/simplex'>simplex</a>, such that <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi><mo stretchy='false'>(</mo><mn>0</mn><mo stretchy='false'>)</mo><mo>=</mo><mi>γ</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo><mo>=</mo><mi>x</mi></mrow><annotation encoding='application/x-tex'>\gamma(0) = \gamma(1) = x</annotation></semantics></math>.</p> <p>A <em>based <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopy</a></em> between two loops is a <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopy</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>Δ</mi> <mn>1</mn></msup></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><mo stretchy='false'>(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo></mrow></mpadded></msup></mtd> <mtd><msup><mo>↘</mo> <mpadded width='0'><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>Δ</mi> <mn>1</mn></msup><mo>×</mo><msup><mi>Δ</mi> <mn>1</mn></msup></mtd> <mtd><mover><mo>→</mo><mi>η</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↑</mo> <mpadded width='0'><mrow><mo stretchy='false'>(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo></mrow></mpadded></msup></mtd> <mtd><msub><mo>↗</mo> <mpadded width='0'><mi>g</mi></mpadded></msub></mtd></mtr> <mtr><mtd><msup><mi>Δ</mi> <mn>1</mn></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ \Delta^1 \\ \downarrow^{\mathrlap{(id,\delta_0)}} & \searrow^{\mathrlap{f}} \\ \Delta^1 \times \Delta^1 &\stackrel{\eta}{\to}& X \\ \uparrow^{\mathrlap{(id,\delta_1)}} & \nearrow_{\mathrlap{g}} \\ \Delta^1 } </annotation></semantics></math></div> <p>such that <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mo stretchy='false'>(</mo><mn>0</mn><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>=</mo><mi>η</mi><mo stretchy='false'>(</mo><mn>1</mn><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>=</mo><mi>x</mi></mrow><annotation encoding='application/x-tex'>\eta(0,-) = \eta(1,-) = x</annotation></semantics></math>.</p> </div> <div class='num_prop'> <h6 id='proposition'>Proposition</h6> <p>This notion of based homotopy is an <a class='existingWikiWord' href='/nlab/show/diff/equivalence+relation'>equivalence relation</a>.</p> </div> <div class='proof'> <h6 id='proof'>Proof</h6> <p>This is directly checked. It is also a special case of the general discussion at <em><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopy</a></em>.</p> </div> <div class='num_definition' id='Concatenation'> <h6 id='definition_3'>Definition</h6> <p>Given two loops <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>γ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>γ</mi> <mn>2</mn></msub><mo>:</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>\gamma_1, \gamma_2 : \Delta^1 \to X</annotation></semantics></math>, define their <strong>concatenation</strong> to be the loop</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>γ</mi> <mn>2</mn></msub><mo>⋅</mo><msub><mi>γ</mi> <mn>1</mn></msub><mo>:</mo><mi>t</mi><mo>↦</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>γ</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mn>2</mn><mi>t</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo stretchy='false'>(</mo><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><msub><mi>γ</mi> <mn>2</mn></msub><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd> <mtd><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>1</mn><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \gamma_2 \cdot \gamma_1 : t \mapsto \left\{ \array{ \gamma_1(2 t) & ( 0 \leq t \leq 1/2 ) \\ \gamma_2(2 (t-1/2)) & (1/2 \leq t \leq 1) } \right. \,. </annotation></semantics></math></div></div> <div class='num_prop' id='GroupStructure'> <h6 id='proposition_2'>Proposition</h6> <p>Concatenation of loops respects based homotopy classes where it becomes an <a class='existingWikiWord' href='/nlab/show/diff/associativity'>associative</a>, <a class='existingWikiWord' href='/nlab/show/diff/identity+element'>unital</a> binary pairing with <a class='existingWikiWord' href='/nlab/show/diff/inverse'>inverses</a>, hence the product in a <a class='existingWikiWord' href='/nlab/show/diff/group'>group</a>.</p> </div> <div class='num_remark'> <h6 id='remark'>Remark</h6> <p>See also at <em><a class='existingWikiWord' href='/nlab/show/diff/path+groupoid'>path groupoid</a></em> for similar constructions.</p> </div> <div class='num_defn'> <h6 id='definition_4'>Definition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> a topological space and <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math> a point, the set of based homotopy equivalence classes of based loops in <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> equipped with the group structure from prop. <a class='maruku-ref' href='#GroupStructure'>2</a> is the <strong>fundamental group</strong> or <strong>first <a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a></strong> of <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_23' 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>, denoted</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_24' 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><mo>∈</mo><mi>Grp</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \pi_1(X,x) \in Grp \,. </annotation></semantics></math></div></div> <p>Hence if we write <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>γ</mi><mo stretchy='false'>]</mo><mo>∈</mo><msub><mi>p</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'>[\gamma] \in p_1(X,x)</annotation></semantics></math> for the based homotopy class of a loop <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math>, then then group operation is</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msub><mi>γ</mi> <mn>1</mn></msub><mo stretchy='false'>]</mo><mo>⋅</mo><mo stretchy='false'>[</mo><msub><mi>γ</mi> <mn>2</mn></msub><mo stretchy='false'>]</mo><mo>≔</mo><mo stretchy='false'>[</mo><msub><mi>γ</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>γ</mi> <mn>2</mn></msub><mo stretchy='false'>]</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> [\gamma_1] \cdot [\gamma_2] \coloneqq [\gamma_1 \cdot \gamma_2] \,. </annotation></semantics></math></div> <div class='num_defn' id='SimplyConnectedSpace'> <h6 id='definition_5'>Definition</h6> <p>A <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> whose fundamental group is trivial is called a <em><a class='existingWikiWord' href='/nlab/show/diff/simply+connected+space'>simply connected topological space</a></em>.</p> </div> <div class='num_defn' id='EMSpace'> <h6 id='definition_6'>Definition</h6> <p>Conversely, a topological space whose only non-trivial <a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a> is the fundamental group is called an <a class='existingWikiWord' href='/nlab/show/diff/Eilenberg-Mac+Lane+space'>Eilenberg-MacLane space</a> denoted <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K(\pi_1(X), 1)</annotation></semantics></math>.</p> </div> <h2 id='properties'>Properties</h2> <h3 id='naturality'>Naturality</h3> <div class='num_prop'> <h6 id='proposition_3'>Proposition</h6> <p>If a topological space <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is path-<a class='existingWikiWord' href='/nlab/show/diff/connected+space'>connected space</a>, then all of the fundamental groups <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_30' 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> are <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphic</a>, for all choices of base points <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math>.</p> </div> <div class='proof'> <h6 id='proof_2'>Proof</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x_0, x_1 \in X</annotation></semantics></math> any two basepoints, there is by assumption a path connecting them, hence a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>:</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'> p : \Delta^1 \to X </annotation></semantics></math></div> <p>such that <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo stretchy='false'>(</mo><mn>0</mn><mo stretchy='false'>)</mo><mo>=</mo><msub><mi>x</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>p(0) = x_0</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo><mo>=</mo><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>p(1) = x_1</annotation></semantics></math>.</p> <p>Write <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>p</mi><mo stretchy='false'>¯</mo></mover><mo>≔</mo><mo stretchy='false'>(</mo><mi>p</mi><mo stretchy='false'>(</mo><mn>1</mn><mo>−</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\bar p \coloneqq (p(1-(-)))</annotation></semantics></math> for the same path with the orientation reversed. Then for <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>γ</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>\gamma_0</annotation></semantics></math> any loop based at <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>x_0</annotation></semantics></math>, the concatenation <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>p</mi><mo>⋅</mo><mo stretchy='false'>(</mo><msub><mi>γ</mi> <mn>0</mn></msub><mo>⋅</mo><mover><mi>p</mi><mo stretchy='false'>¯</mo></mover><mo stretchy='false'>)</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'> [ p \cdot (\gamma_0 \cdot \bar p) ]</annotation></semantics></math>, def. <a class='maruku-ref' href='#Concatenation'>1</a> yield a loop based at <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>x_1</annotation></semantics></math> (obtained from <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msub><mi>γ</mi> <mn>0</mn></msub><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[\gamma_0]</annotation></semantics></math> by <a class='existingWikiWord' href='/nlab/show/diff/adjoint+action'>conjugation</a> with <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>p</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[p]</annotation></semantics></math>).</p> <p>It is immediate to check that this induces an <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Ad</mi> <mrow><mo stretchy='false'>[</mo><mi>p</mi><mo stretchy='false'>]</mo></mrow></msub><mo>:</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><msub><mi>x</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> Ad_{[p]} : \pi_1(X,x_0) \to \pi_1(X,x_1) \,. </annotation></semantics></math></div></div> <div class='num_remark'> <h6 id='remark_2'>Remark</h6> <p>Therefore one sometimes loosely speaks of ‘the’ fundamental group of a connected space. But beware that the isomorphism in the above construction is not unique. Therefore forming fundamental groups is not a <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> on connected spaces.</p> </div> <div class='num_prop'> <h6 id='proposition_4'>Proposition</h6> <p>It is, however, a functor on <a class='existingWikiWord' href='/nlab/show/diff/pointed+object'>pointed topological spaces</a>: <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>:</mo></mrow><annotation encoding='application/x-tex'>\pi_1(-) :</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mrow /> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup><mo>→</mo></mrow><annotation encoding='application/x-tex'>{}^{*/} \to</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/Grp'>Grp</a>.</p> </div> <h3 id='RelationToSingularHomology'>Relation to singular homology</h3> <p>The fundamental group is in general non-abelian (i.e. is not an <a class='existingWikiWord' href='/nlab/show/diff/abelian+group'>abelian group</a>), the <a href='#Examples'>Examples</a> below. For a connected topological space <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, its <em><a class='existingWikiWord' href='/nlab/show/diff/abelianization'>abelianization</a></em> is equivalent to the first <a class='existingWikiWord' href='/nlab/show/diff/singular+homology'>singular homology</a> group</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_47' 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><msup><mo stretchy='false'>)</mo> <mi>ab</mi></msup><mo>≃</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \pi_1(X,x)^{ab} \simeq H_1(X) \,. </annotation></semantics></math></div> <p>See at <em><a href='singular%20homology#RelationToHomotopyGroups'>singular homology – Relation to homotopy groups</a></em> for more on this.</p> <h3 id='relation_to_universal_covers_and_galois_groups'>Relation to universal covers and Galois groups</h3> <p>There is a relation to <a class='existingWikiWord' href='/nlab/show/diff/universal+covering+space'>universal covers</a>: Under suitable conditions the group of cover automorphisms of a universal cover is isomorphic to the fundamental group of the covered space. This is the topic of the <em><a class='existingWikiWord' href='/nlab/show/diff/%C3%A9tale+homotopy'>étale fundamental group</a></em>, also referred to at <em><a class='existingWikiWord' href='/nlab/show/diff/Chevalley+fundamental+group'>Chevalley fundamental group</a></em>, see there for more.</p> <p>In particular in <a class='existingWikiWord' href='/nlab/show/diff/algebraic+geometry'>algebraic geometry</a> and <a class='existingWikiWord' href='/nlab/show/diff/arithmetic+geometry'>arithmetic geometry</a> this essentially identifies the concept of fundamental group with that of <em><a class='existingWikiWord' href='/nlab/show/diff/Galois+group'>Galois groups</a></em>. For this reason one also speaks of the <em><a class='existingWikiWord' href='/nlab/show/diff/algebraic+fundamental+group'>algebraic fundamental group</a></em> in this context. See at <em><a class='existingWikiWord' href='/nlab/show/diff/Galois+theory'>Galois theory</a></em> for more on this.</p> <p>See also at <em><span class='newWikiWord'>link between Galois theory and fundamental groups<a href='/nlab/new/link+between+Galois+theory+and+fundamental+groups'>?</a></span></em>.</p> <p>In <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck%27s+Galois+theory'>Grothendieck's Galois theory</a>, the role of the basepoint is replaced by considering a ‘fibre functor’ <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>:</mo><mi>𝒞</mi><mo>→</mo><mi>Sets</mi></mrow><annotation encoding='application/x-tex'>F:\mathcal{C}\to Sets</annotation></semantics></math> or to <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>FinSets</mi></mrow><annotation encoding='application/x-tex'>FinSets</annotation></semantics></math>, where <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> is the category of coverings of the given space. This theory extends to other situations and the term <a class='existingWikiWord' href='/nlab/show/diff/algebraic+fundamental+group'>algebraic fundamental group</a> is used in particular for the case of <a class='existingWikiWord' href='/nlab/show/diff/scheme'>scheme</a>s (of a suitable type); see (SGA1).</p> <h2 id='generalizations'>Generalizations</h2> <h3 id='nonlocally_nice_spaces_and_generalised_spaces'>Non-locally ‘nice’ spaces and ‘generalised’ spaces</h3> <p>The definition of fundamental group in terms of homotopy classes of loops at a base point does not work well for the spaces that occur in <a class='existingWikiWord' href='/nlab/show/diff/algebraic+geometry'>algebraic geometry</a>, nor for many spaces considered in analysis as there may be very few loops. For instance, for a <a class='existingWikiWord' href='/nlab/show/diff/scheme'>scheme</a> there are in general very few paths, and <a class='existingWikiWord' href='/nlab/show/diff/Alexander+Grothendieck'>Grothendieck</a> gave a definition of a fundamental group in SGA1 which is closely related to the <a class='existingWikiWord' href='/nlab/show/diff/Galois+group'>Galois groups</a> of number theory, but in cases where both the path-based group and this <a class='existingWikiWord' href='/nlab/show/diff/algebraic+fundamental+group'>algebraic fundamental group</a> make sense, the algebraic form tends to be related to the <a class='existingWikiWord' href='/nlab/show/diff/profinite+completion+of+a+group'>profinite completion</a> of the topological fundamental group; see the example in that entry.</p> <p>A similar type of construction gives the <a class='existingWikiWord' href='/nlab/show/diff/fundamental+group+of+a+topos'>fundamental group of a topos</a>. Other related forms include a Čech version of the fundamental group used in shape theory, and linked to Čech homology groups of a compact space.</p> <p>The notion of fundamental group generalizes to that of <a class='existingWikiWord' href='/nlab/show/diff/fundamental+groupoid'>fundamental groupoid</a> in both the loop based theory and in <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck%27s+Galois+theory'>Grothendieck's Galois theory</a> as described in <a class='existingWikiWord' href='/nlab/show/diff/SGA1'>SGA1</a>. In this form it has been used to give generalisations for simplicial profinite spaces in work by Quick and to <a class='existingWikiWord' href='/nlab/show/diff/pro-space'>pro-spaces</a> in work of Isaksen.</p> <h3 id='proper_fundamental_groups'>Proper fundamental groups</h3> <p>In the context of <a class='existingWikiWord' href='/nlab/show/diff/proper+homotopy+theory'>proper homotopy theory</a> there are two related fundamental groups for single ended spaces.</p> <h2 id='Examples'>Examples</h2> <div class='num_example' id='EuclideanSpaceFundamentalGroup'> <h6 id='example'>Example</h6> <p><strong>(Euclidean space is <a class='existingWikiWord' href='/nlab/show/diff/simply+connected+space'>simply connected</a>)</strong></p> <p>For <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_51' 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_664d13078f5b982a7b1b637781723d93218ada08_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^n</annotation></semantics></math> be the <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-dimensional <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a> with its <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>. Then for every point <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>x \in \mathbb{R}^n</annotation></semantics></math> the fundamental group is trivial:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>=</mo><mn>1</mn><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \pi_1(\mathbb{R}^n, x) = 1 \,. </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof_3'>Proof</h6> <p>Let</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>⟶</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'> \gamma \;\colon\; [0,1] \longrightarrow \mathbb{R}^n </annotation></semantics></math></div> <p>be <a class='existingWikiWord' href='/nlab/show/diff/loop'>loop</a> at <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>, hence a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> with <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi><mo stretchy='false'>(</mo><mn>0</mn><mo stretchy='false'>)</mo><mo>=</mo><mi>x</mi></mrow><annotation encoding='application/x-tex'>\gamma(0) = x</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo><mo>=</mo><mi>x</mi></mrow><annotation encoding='application/x-tex'>\gamma(1) = x</annotation></semantics></math>. Using the <a class='existingWikiWord' href='/nlab/show/diff/real+vector+space'>real vector space</a> structure on <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^n</annotation></semantics></math>, we may define the function</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><msup><mi>ℝ</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><mo stretchy='false'>(</mo><mi>t</mi><mo>,</mo><mi>s</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>x</mi><mo>+</mo><mi>s</mi><mo stretchy='false'>(</mo><mi>γ</mi><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo>−</mo><mi>x</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ [0,1] \times [0,1] &\overset{\eta}{\longrightarrow}& \mathbb{R}^n \\ (t,s) &\mapsto& x + s (\gamma(t) - x) } \,. </annotation></semantics></math></div> <p>This is a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a>, since it is the composite of the continuous function <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>id</mi> <mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow></msub><mo>×</mo><mi>γ</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><mi>t</mi><mo>,</mo><mi>s</mi><mo stretchy='false'>)</mo><mo>↦</mo><mo stretchy='false'>(</mo><mi>γ</mi><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>s</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(id_{[0,1]} \times \gamma) \;\colon\; (t,s) \mapsto (\gamma(t),s)</annotation></semantics></math> (which is continuous as the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product</a> of two continuous functions) and the function <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>v</mi><mo>,</mo><mi>s</mi><mo stretchy='false'>)</mo><mo>↦</mo><mi>x</mi><mo>+</mo><mi>s</mi><mo stretchy='false'>(</mo><mi>v</mi><mo>−</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(v,s) \mapsto x + s ( v - x )</annotation></semantics></math> (which is continuous since <a class='existingWikiWord' href='/nlab/show/diff/real+polynomial+function'>polynomials are continuous</a>).</p> <p>Moreover, by construction we have</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo><mo>=</mo><mi>γ</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mphantom><mi>AAAA</mi></mphantom><mi>η</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mn>0</mn><mo stretchy='false'>)</mo><mo>=</mo><msub><mi>const</mi> <mi>x</mi></msub><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \eta(-,1) = \gamma(-) \phantom{AAAA} \eta(-,0) = const_x \,. </annotation></semantics></math></div> <p>Therefore this is a <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopy</a> from <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi></mrow><annotation encoding='application/x-tex'>\gamma</annotation></semantics></math> to the constant loop at <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>.</p> </div> <div class='num_example'> <h6 id='example_2'>Example</h6> <p>By definition <a class='maruku-ref' href='#SimplyConnectedSpace'>3</a>, the fundamental group of every <a class='existingWikiWord' href='/nlab/show/diff/simply+connected+space'>simply connected topological space</a> is trivial.</p> </div> <div class='num_example'> <h6 id='example_3'>Example</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/fundamental+group+of+the+circle+is+the+integers'>fundamental group of the circle is the integers</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_67' 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' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \pi_1(S^1) \simeq \mathbb{Z} \,. </annotation></semantics></math></div></div> <div class='num_remark'> <h6 id='remark_3'>Remark</h6> <p>An instructive formalization of this basic statement in <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a> is in (<a href='#Shulman'>Shulman</a>).</p> </div> <div class='num_example'> <h6 id='example_4'>Example</h6> <p>By definition <a class='maruku-ref' href='#EMSpace'>4</a>, the fundamental group of any <a class='existingWikiWord' href='/nlab/show/diff/Eilenberg-Mac+Lane+space'>Eilenberg-MacLane space</a> <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mi>G</mi><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K(G,1)</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>: <math class='maruku-mathml' display='inline' id='mathml_664d13078f5b982a7b1b637781723d93218ada08_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>K</mi><mo stretchy='false'>(</mo><mi>G</mi><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>=</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>\pi_1(K(G,1)) = G</annotation></semantics></math>.</p> </div> <h2 id='related_concept'>Related concept</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a></p> </li> <li> <p><a href='smooth+homotopy#HomotopyOfSmoothPathsRelativeToTheirEndpoints'>homotopy of smooth paths relative to their endpoints</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/winding+number'>winding number</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebraic+fundamental+group'>algebraic fundamental group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/anabelian+geometry'>anabelian geometry</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Chevalley+fundamental+group'>Chevalley fundamental group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%C3%A9tale+homotopy'>etale homotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck%27s+Galois+theory'>Grothendieck's Galois theory</a></p> </li> </ul> <h2 id='references'>References</h2> <p>Historical origins:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Henri+Poincar%C3%A9'>Henri Poincaré</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Analysis+Situs'>Analysis Situs</a></em>, Journal de l’École Polytechnique. (2). 1: 1–123 (1895) (Orig.: <a href='https://gallica.bnf.fr/ark:/12148/bpt6k4337198/f7'>gallica:12148/bpt6k4337198/f7</a>, Engl.: <a class='existingWikiWord' href='/nlab/show/diff/John+Stillwell'>John Stillwell</a>: <a href='https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/Stillwell_AnalysisSitus.pdf' title='pdf'>pdf</a>)</li> </ul> <p>Textbook accounts:</p> <ul> <li id='AGP02'><a class='existingWikiWord' href='/nlab/show/diff/Marcelo+Aguilar'>Marcelo Aguilar</a>, <a class='existingWikiWord' href='/nlab/show/diff/Samuel+Gitler'>Samuel Gitler</a>, <a class='existingWikiWord' href='/nlab/show/diff/Carlos+Prieto'>Carlos Prieto</a>, Section 2.5 of: <em>Algebraic topology from a homotopical viewpoint</em>, Springer (2002) (<a href='http://tocs.ulb.tu-darmstadt.de/106999419.pdf'>toc pdf</a>)</li> </ul> <p>Review and Exposition:</p> <ul> <li id='Moller11'> <p><a class='existingWikiWord' href='/nlab/show/diff/Jesper+Michael+M%C3%B8ller'>Jesper Møller</a>, <em>The fundamental group and covering spaces</em><span> (2011)<del class='diffmod'> [[pdf](http://www.math.ku.dk/~moller/f03/algtop/notes/covering.pdf),</del><ins class='diffmod'> [[arXiv:1106.5650](https://arxiv.org/abs/1106.5650),</ins></span><ins class='diffins'><a href='http://www.math.ku.dk/~moller/f03/algtop/notes/covering.pdf'>pdf</a></ins><ins class='diffins'>, </ins><a class='existingWikiWord' href='/nlab/files/Moller-FundamentalGroup.pdf' title='pdf'>pdf</a>]</p> </li> <li> <p>Alberto Santini, <em>Topological groupoids</em> (2011) [[pdf](http://web.math.ku.dk/~moller/students/alberto.pdf), <a class='existingWikiWord' href='/nlab/files/Santini-Groupoids.pdf' title='pdf'>pdf</a>]</p> <blockquote> <p>(about <a class='existingWikiWord' href='/nlab/show/diff/groupoid'>groupoids</a> in <a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a>, notably <a class='existingWikiWord' href='/nlab/show/diff/fundamental+groupoid'>fundamental groupoids</a> – not about <a class='existingWikiWord' href='/nlab/show/diff/topological+groupoid'>topological groupoids</a>)</p> </blockquote> </li> <li> <p><em><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology+--+2'>Introduction to Topology -- 2</a></em></p> </li> </ul> <p>Discussion from the point of view of <a class='existingWikiWord' href='/nlab/show/diff/Galois+theory'>Galois theory</a> is in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Luis+Javier+Hern%C3%A1ndez+Paricio'>Luis Javier Hernández-Paricio</a>, <em>Fundamental pro-groupoids and covering projections</em>(<a href='http://matwbn.icm.edu.pl/ksiazki/fm/fm156/fm15611.pdf'>pdf</a>)</li> </ul> <p>See also:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Isaksen'>D. C. Isaksen</a>, <em>A model structure on the category of pro-simplicial sets</em>, Trans. Amer. Math. Soc., 353, (2001), 2805–2841</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Gereon+Quick'>G. Quick</a>, <em>Profinite homotopy theory</em>, Documenta Mathematica, 13, (2008), 585–612.</p> </li> </ul> <p>Proof in <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a> that the <a class='existingWikiWord' href='/nlab/show/diff/fundamental+group+of+the+circle+is+the+integers'>fundamental group of the circle is the integers</a>:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Licata'>Daniel Licata</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mike+Shulman'>Michael Shulman</a>, <em>Calculating the Fundamental Group of the Circle in Homotopy Type Theory</em>, (<a href='https://arxiv.org/abs/1301.3443'>arXiv:1301.3443</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/UF-IAS-2012'>Univalent Foundations Project</a>, section 8.1 of <em><a class='existingWikiWord' href='/nlab/show/diff/Homotopy+Type+Theory+--+Univalent+Foundations+of+Mathematics'>Homotopy Type Theory -- Univalent Foundations of Mathematics</a></em></p> </li> </ul> <p>the HoTT-<a class='existingWikiWord' href='/nlab/show/diff/Coq'>Coq</a>-code is at</p> <ul> <li id='Shulman'><a class='existingWikiWord' href='/nlab/show/diff/Mike+Shulman'>Mike Shulman</a>, <em><a href='https://github.com/HoTT/HoTT/blob/master/Coq/HIT/Pi1S1.v'>P1S1.v</a></em></li> </ul> <p> </p> </div> <div class="revisedby"> <p> Last revised on January 18, 2023 at 13:42:38. 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