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href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> <h4 id="topos_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a></strong></p> <h2 id="sidebar_background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf">(∞,1)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> <h2 id="sidebar_definitions">Definitions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elementary+%28%E2%88%9E%2C1%29-topos">elementary (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">localization of an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+localization">topological localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercompletion">hypercompletion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>/<a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-topos">(n,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/n-topos">n-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/truncated">n-truncated object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected">n-connected object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">(1,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-presheaf">(2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-quasitopos">(∞,1)-quasitopos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+%28%E2%88%9E%2C1%29-presheaf">separated (∞,1)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+presheaf">separated presheaf</a></li> </ul> </li> <li> <p><span class="newWikiWord">(2,1)-quasitopos<a href="/nlab/new/%282%2C1%29-quasitopos">?</a></span></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+%282%2C1%29-presheaf">separated (2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-topos">(∞,2)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-topos">(∞,n)-topos</a></p> </li> </ul> <h2 id="sidebar_characterization">Characterization</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+colimits">universal colimits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object in an (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/effective+epimorphism">effective epimorphism</a></li> </ul> </li> </ul> <h2 id="sidebar_morphisms">Morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-geometric+morphism">(∞,1)-geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Topos">(∞,1)Topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lawvere+distribution">Lawvere distribution</a></p> </li> </ul> <h2 id="sidebar_extra">Extra stuff, structure and property</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+%28%E2%88%9E%2C1%29-topos">hypercomplete (∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+object">hypercomplete object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+theorem">Whitehead theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over-%28%E2%88%9E%2C1%29-topos">over-(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-localic+%28%E2%88%9E%2C1%29-topos">n-localic (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+n-connected+%28n%2C1%29-topos">locally n-connected (n,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+%28%E2%88%9E%2C1%29-topos">local (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/concrete+%28%E2%88%9E%2C1%29-sheaf">concrete (∞,1)-sheaf</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></p> </li> </ul> <h2 id="sidebar_models">Models</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</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/model+structure+on+functors">model structure on functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+site">model site</a>/<a class="existingWikiWord" href="/nlab/show/sSet-site">sSet-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent+for+simplicial+presheaves">descent for simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Verity+on+descent+for+strict+omega-groupoid+valued+presheaves">descent for presheaves with values in strict ∞-groupoids</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_constructions">Constructions</h2> <p><strong>structures in a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/shape+of+an+%28%E2%88%9E%2C1%29-topos">shape</a> / <a class="existingWikiWord" href="/nlab/show/coshape+of+an+%28%E2%88%9E%2C1%29-topos">coshape</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a>/<a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical</a>/<a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric</a> homotopy groups</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Postnikov+tower+in+an+%28%E2%88%9E%2C1%29-category">Postnikov tower</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+tower+in+an+%28%E2%88%9E%2C1%29-topos">Whitehead tower</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory+in+an+%28%E2%88%9E%2C1%29-topos">rational homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+dimension">homotopy dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomological+dimension">cohomological dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covering+dimension">covering dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heyting+dimension">Heyting dimension</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/%28infinity%2C1%29-topos+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#statement'>Statement</a></li> <ul> <li><a href='#traditional'>Traditional</a></li> <li><a href='#in_higher_topos_theory'>In higher topos theory</a></li> <li><a href='#HigherCubical'>Higher cubical BM-theorems and analytic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-functors</a></li> </ul> <li><a href='#references'>References</a></li> <ul> <li><a href='#classical'>Classical</a></li> <li><a href='#ReferencesInHoTT'>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-topos theory and homotopy type theory</a></li> <li><a href='#in_shape_theory'>In shape theory</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>Blakers-Massey theorem</em> in the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a> is concerned with algebraically describing the first obstruction to <a class="existingWikiWord" href="/nlab/show/excision">excision</a> for relative <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a>. There is also a weaker version just describing vanishing conditions which should be called the <em>Blakers-Massey connectivity theorem</em>.</p> <h2 id="statement">Statement</h2> <h3 id="traditional">Traditional</h3> <p>This <a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a> is measured by triad homotopy groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>m</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_m(X;A,B)</annotation></semantics></math> for a pointed space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with two subspaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A,B</annotation></semantics></math> each containing the base point. Here the group structure is defined for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>≥</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">m \geq 3</annotation></semantics></math> and is abelian for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>≥</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">m \geq 4</annotation></semantics></math>. There is an exact sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo><msub><mi>π</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo><msup><mo>→</mo> <mi>ϵ</mi></msup><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex">\cdots \to \pi_{n+1}(X;A,B) \to \pi_n (A, A \cap B) \to^{\epsilon} \pi_n( X,B) \to \pi_n(X;A,B) \to \cdots </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> is the excision map. The main result of Blakers and Massey is as follows:</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>Suppose the triad <strong>X</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>=</mo><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> =(X;A,B)</annotation></semantics></math> is such that: (i) the interiors of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A,B</annotation></semantics></math> cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>; (ii) that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A,B</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mi>A</mi><mo>∩</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">C=A \cap B</annotation></semantics></math> are connected; (iii) that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is simply connected; (iv) and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,C)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(m-1)</annotation></semantics></math>-connected and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(B,C)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mo>≥</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">m,n \geq 3</annotation></semantics></math>. Then <strong>X</strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>=</mo><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">=(X;A,B)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(m+n-2)</annotation></semantics></math>-connected and if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is simply connected then the morphism given by the generalised Whitehead product</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>m</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>⊗</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mrow><mi>m</mi><mo>+</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \pi_{m}(A,C) \otimes \pi_{n}(B,C) \to \pi_{m+n-1}(X;A,B)</annotation></semantics></math></div> <p><em>is an isomorphism</em>.</p> </div> <p>(<a href="#BlakersMassey51">Blakers-Massey 51</a>, <a href="#tomDieck08">tomDieck 08, theorem 6.4.1</a>).</p> <div class="num_remark" id="InTermsOfPushouts"> <h6 id="remark">Remark</h6> <p>A more intrinsic statement in the language of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of the connectivity part of the theorem is that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">f_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f_2</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/maps">maps</a> out of the same <a class="existingWikiWord" href="/nlab/show/domain">domain</a> which are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">n_1</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/n-connected+morphism">connective</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">n_2</annotation></semantics></math>-connective, respectively, then the canonical map from that domain into the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> of their <a class="existingWikiWord" href="/nlab/show/homotopy+pushout">homotopy pushout</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msub><mi>Y</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><msub><mi>Y</mi> <mn>2</mn></msub></mtd> <mtd></mtd> <mtd><msub><mi>Y</mi> <mn>2</mn></msub><munder><mo>×</mo><mrow><msub><mi>Y</mi> <mn>2</mn></msub><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>X</mi></munder><msub><mi>Y</mi> <mn>1</mn></msub></mrow></munder><msub><mi>Y</mi> <mn>1</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &\stackrel{f_1}{\longrightarrow}& Y_1 \\ \downarrow^{\mathrlap{f_2}} & \searrow \\ Y_2 && Y_2 \underset{Y_2 \underset{X}{\coprod} Y_1}{\times} Y_1 } </annotation></semantics></math></div> <p>is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n_1 + n_2 - 1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/n-connected+morphism">connective</a>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>For the special case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mn>1</mn></msub><mo>≃</mo><msub><mi>Y</mi> <mn>2</mn></msub><mo>≃</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">Y_1 \simeq Y_2 \simeq \ast</annotation></semantics></math> are point contractible, the Blakers-Massey theorem reduces to the <a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a>.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>Since the tensor product is zero if one of its factors is zero, this result also gives criteria for the excision morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> to be an isomorphism in a certain range of dimensions. For this reason the excision consequences of that sequence are also called the <em>excision theorem of Blakers and Massey</em> and have been given quite separate proofs for example in (<a href="#Hatcher">Hatcher</a>), and in (<a href="#tomDieck08">tom Dieck</a>). The first non zero triad homotopy group is also called the <em>critical group</em>. Note that in <em>algebraic topology</em> one wants <em>algebraic</em> results, not just connectivity results.</p> </div> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>A natural question is what happens if the conditions that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mo>≥</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">m,n \geq 3</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> simply connected are weakened. For example in the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>=</mo><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">m=n=2</annotation></semantics></math> we have the additional structure that the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_2(A,C) \to \pi_1(C), \pi_2(B,C) \to \pi_1(C)</annotation></semantics></math> are crossed modules, and so the required relative homotopy groups are in general nonabelian. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>≥</mo><mn>3</mn><mo>,</mo><mi>n</mi><mo>≥</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">m \geq 3 ,n \geq 3</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>m</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_m(A,C), \pi_n(B,C)</annotation></semantics></math> are still <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(C)</annotation></semantics></math>-modules.</p> <p>The extension to the non simply connected case was given by Brown and Loday; one simply replaces the usual tensor product by the nonabelian tensor product of groups which act on each other and on themselves by conjugation. This result is a special case of a Seifert-van Kampen Theorem for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cubes of spaces. Notice that the assumption (i) of the theorem is reminiscent of such a type of theorem. The useful fact is that one gets such a theorem for a certain kind of <em>structured space</em> which allows for the development of algebraic structures which have structures in a range of dimensions.</p> <p>Thus one of the intuitions is that the Blakers-Massey Theorem, and hence also the FST, is of the Seifert-van Kampen type, since we are assuming that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the union of the interiors of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A,B</annotation></semantics></math>.</p> </div> <h3 id="in_higher_topos_theory">In higher topos theory</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>The Blakers-Massey connectivity theorem in the form of remark <a class="maruku-ref" href="#InTermsOfPushouts"></a> holds in every <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaves">(∞,1)-sheaves</a>.</p> </div> <p>This is shown in (<a href="#Rezk10">Rezk 10, prop. 8.16</a>) with reference to <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sites">(∞,1)-sites</a>. An intrinsic proof in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> is announced in (<a href="#HoTTBook">HoTTBook, theorem 8.10.2</a>, <a href="#LumsdaineFinsterLicata13">Lumsdaine-Finster-Licata 13</a>). The fully formal computer-checked version of this proof appears as HoTT-<a class="existingWikiWord" href="/nlab/show/Agda">Agda</a> code in (<a href="#Favonia">Favonia 14</a>).</p> <p>This translates to an <a class="existingWikiWord" href="/nlab/show/internal+language">internal language</a> proof of Blakers-Massey valid in all <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-toposes">(∞,1)-toposes</a> (including <a class="existingWikiWord" href="/nlab/show/elementary+%28%E2%88%9E%2C1%29-toposes">elementary (∞,1)-toposes</a>). Unwinding of the fully formal HoTT proof to ordinary mathematical language is, for the special case of the <a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a>, in (<a href="#Rezk14">Rezk 14</a>).</p> <h3 id="HigherCubical">Higher cubical BM-theorems and analytic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-functors</h3> <p>There are higher analogs of the BM-theorem with (pushout) squares replaced by higher dimensional cubes. The higher BM-theorem (<a href="#Goodwillie91">Goodwillie 91</a>) says equivalently that the identity <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> on <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> is a 1-<a class="existingWikiWord" href="/nlab/show/analytic+%28%E2%88%9E%2C1%29-functor">analytic (∞,1)-functor</a>. See (<a href="#MunsonVolic15">Munson-Volic 15, section 6</a>).</p> <h2 id="references">References</h2> <h3 id="classical">Classical</h3> <p>The original connectivity statement:</p> <ul> <li id="BlakersMassey51"><a class="existingWikiWord" href="/nlab/show/Albert+Blakers">Albert Blakers</a>, <a class="existingWikiWord" href="/nlab/show/William+Massey">William Massey</a>, <em>The homotopy groups of a triad I</em> , Annals of Mathematics 53: 161–204, (1951) (<a href="https://www.jstor.org/stable/1969346">jstor:1969346</a>)</li> </ul> <p>The algebraic statement and proof:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Albert+Blakers">Albert Blakers</a>, <a class="existingWikiWord" href="/nlab/show/William+Massey">William Massey</a>, <em>The homotopy groups of a triad. {III}</em>, Ann. of Math. (2), 58: (1953) 409–417 (<a href="https://www.jstor.org/stable/1969744">jstor:1969744</a>)</li> </ul> <p>Textbook accounts:</p> <ul> <li id="tDKP70"> <p><a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Heiner+Kamps">Klaus Heiner Kamps</a>, <a class="existingWikiWord" href="/nlab/show/Dieter+Puppe">Dieter Puppe</a>, Section 15 of: <em>Homotopietheorie</em>, Lecture Notes in Mathematics <strong>157</strong> Springer 1970 (<a href="https://link.springer.com/book/10.1007/BFb0059721">doi:10.1007/BFb0059721</a>)</p> </li> <li id="Kochmann96"> <p><a class="existingWikiWord" href="/nlab/show/Stanley+Kochmann">Stanley Kochmann</a>, theorem 3.2.4 of <em><a class="existingWikiWord" href="/nlab/show/Bordism%2C+Stable+Homotopy+and+Adams+Spectral+Sequences">Bordism, Stable Homotopy and Adams Spectral Sequences</a></em>, AMS 1996</p> </li> <li id="Hatcher"> <p><a class="existingWikiWord" href="/nlab/show/Alan+Hatcher">Alan Hatcher</a>, theorem 4.23 <em><a href="http://www.math.cornell.edu/~hatcher/AT/ATpage.html">Algebraic Topology</a></em></p> </li> <li id="tomDieck08"> <p><a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, theorem 6.4.1 <em>Algebraic Topology</em>, EMS Textbooks in Mathematics, (2008) (<a href="http://www.maths.ed.ac.uk/~aar/papers/diecktop.pdf">pdf</a>)</p> </li> </ul> <p>The Blakers-Massey’s Connectivity Theorem was shown to be a consequence of Farjoun’s “cellular inequalities”</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Emmanuel+Dror+Farjoun">Emmanuel Dror Farjoun</a>, <em>Cellular spaces, null spaces and homotopy localization</em>m No. 1621-1622. Springer, 1996]</li> </ul> <p>is Theorem 1.B of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Wojciech+Chach%C3%B3lski">Wojciech Chachólski</a>, <em>A generalization of the triad theorem of Blakers-Massey</em> Topology 36.6 (1997): 1381-1400</li> </ul> <p>This would constitute a purely <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy-theoretic</a> proof.</p> <p>The generalisation of the algebraic statement is Theorem 4.3 in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ronnie+Brown">R. Brown</a> and <a class="existingWikiWord" href="/nlab/show/Jean-Louis+Loday">Jean-Louis Loday</a>, Homotopical excision, and Hurewicz theorems, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cubes of spaces, <em>Proc. London Math. Soc.</em> <p>(3) 54 (1987) 176-192. <a href="https://groupoids.org.uk/pdffiles/VKTEVANS2.pdf">pdf</a></p> </li> </ul> <p>which relies essentially on the paper</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ronnie+Brown"> R. Brown</a> and J.-L. Loday, Van Kampen theorems for diagrams of <p>spaces, <em>Topology</em> 26 (1987) 311-334,</p> </li> </ul> <p>for the van Kampen Theorem and for the nonabelian tensor product of groups. Here is a link to a bibliography of 170 items on the <a href="http://groupoids.org.uk/nonabtens.html">nonabelian tensor product</a>.</p> <p>Further applications are explained in</p> <p><a class="existingWikiWord" href="/nlab/show/Ronnie+Brown">R. Brown</a>, Triadic Van Kampen theorems and Hurewicz theorems, _</p> <p>Algebraic Topology, Proc. Int. Conf. March 1988_, Edited M.Mahowald and S.Priddy, Cont. Math. 96 (1989) 39-57. <a href="http://groupoids.org.uk//pdffiles/VKTEVANS2.pdf">pdf</a></p> <p>The following paper applies the methods of the above two Brown-Loday papers to the well known problem of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-ad connectivity and to determination of the critical group, see Theorem 3.8 of:</p> <ul> <li>Ellis, G.J. and Steiner, R. Higher-dimensional crossed modules and the homotopy groups of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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>-ads. <em>J. Pure Appl. Algebra</em> 46 (1987) 117–136.</li> </ul> <p>The methods work because of their equivalence between cat<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo></mo><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">^n</annotation></semantics></math>-groups and crossed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cubes of groups. This can be explained by saying that we need two kinds of algebraic categories for calculations with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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>-types: <em>broad</em> categories for conjecturing and proving theorems, and <em>narrow</em> algebraic categories for calculations and relations with classical ideas. In this case the broad category is that of cat<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo></mo><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">^n</annotation></semantics></math>-groups, and the narrow category is that of crossed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cubes of groups, which are related geometrically to the homotopy groups of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math>-ads and to generalised Whitehead products. The tricky equivalence between the two kinds of categories is one of the engines behind the results, since it enables the use of whichever category is most convenient at any given time. Note also these two categories model weak, pointed, homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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>-types, as shown by Loday in his paper</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jean-Louis+Loday">Jean-Louis Loday</a>, Spaces with finitely many non-trivial homotopy groups, <em>J. Pure Appl. Algebra</em> 24 (1982) 179-202.</li> </ul> <p>Further background to these ideas is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ronnie+Brown"> R. Brown</a>“A philosophy of modelling and computing homotopy types” Presentation to CT2015, Aveiro, Portugal, June 14-19. <a href="https://groupoids.org.uk//pdffiles/aveiro-beamer-handout.pdf">pdf</a> and in “Modelling and computing homotopy types: I” Indag.Math. 29 (2018) 459-482 <a href="https://www.sciencedirect.com/science/article/pii/S0019357717300460">pdf</a></li> </ul> <p>Discussion of Blakers-Massey connectivity for <a class="existingWikiWord" href="/nlab/show/ring+spectra">ring spectra</a>/<a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+rings">E-∞ rings</a> and other <a class="existingWikiWord" href="/nlab/show/algebras+over+operads">algebras over operads</a> is in</p> <ul> <li>Michael Ching, <a class="existingWikiWord" href="/nlab/show/John+Harper">John Harper</a>, <em>Higher homotopy excision and Blakers-Massey theorems for structured ring spectra</em> (<a href="http://arxiv.org/abs/1402.4775">arXiv:1402.4775</a>)</li> </ul> <p>The higher cubical version of Blakers-Massey connectivity is due to</p> <ul> <li id="Goodwillie91"><a class="existingWikiWord" href="/nlab/show/Tom+Goodwillie">Tom Goodwillie</a>, <em>Calculus II: Analytic functors</em>, K-Theory 01/1991; 5(4):295-332. DOI: 10.1007/BF00535644</li> </ul> <p>a textbook account is in</p> <ul> <li id="MunsonVolic15"><a class="existingWikiWord" href="/nlab/show/Brian+Munson">Brian Munson</a>, <a class="existingWikiWord" href="/nlab/show/Ismar+Volic">Ismar Volic</a>, <em>Cubical homotopy theory</em>, Cambridge University Press, 2015 <a href="http://palmer.wellesley.edu/~ivolic/pdf/Papers/CubicalHomotopyTheory.pdf">pdf</a></li> </ul> <h3 id="ReferencesInHoTT">In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-topos theory and homotopy type theory</h3> <p>A proof of Blakers-Massey connectivity in general <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-toposes">(∞,1)-toposes</a> is in prop. 8.16 of</p> <ul> <li id="Rezk10"><a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, <em>Toposes and homotopy toposes</em> (2010) (<a href="http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf">pdf</a>)</li> </ul> <p>A general version of the connectivity theorem in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> (and thus in <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos+theory">(infinity,1)-topos theory</a>) was found by</p> <ul> <li id="LumsdaineFinsterLicata13"><a class="existingWikiWord" href="/nlab/show/Peter+LeFanu+Lumsdaine">Peter LeFanu Lumsdaine</a>, <a class="existingWikiWord" href="/nlab/show/Eric+Finster">Eric Finster</a>, <a class="existingWikiWord" href="/nlab/show/Dan+Licata">Dan Licata</a> (to appear)</li> </ul> <p>A fully computer-checked version of this proof in HoTT-<a class="existingWikiWord" href="/nlab/show/Agda">Agda</a> was produced in</p> <ul> <li id="Favonia"><a class="existingWikiWord" href="/nlab/show/Favonia">Favonia</a>, <em><a href="https://github.com/HoTT/HoTT-Agda/blob/1.0/Homotopy/BlakersMassey.agda">BlakersMassey.agda</a></em></li> </ul> <p>the statement appeared also as</p> <ul> <li id="HoTTBook"><a class="existingWikiWord" href="/nlab/show/Univalent+Foundations+Project">Univalent Foundations Project</a>, theorem 8.10.2 of <em><a class="existingWikiWord" href="/nlab/show/Homotopy+Type+Theory+--+Univalent+Foundations+of+Mathematics">Homotopy Type Theory – Univalent Foundations of Mathematics</a></em></li> </ul> <p>and an announcement was given in</p> <ul> <li id="Lumsdaine13"><a class="existingWikiWord" href="/nlab/show/Peter+LeFanu+Lumsdaine">Peter LeFanu Lumsdaine</a>, <em>The Blakers-Massey theorem in homotopy type theory</em> talk at <a href="http://www.crm.cat/en/Activities/Pages/ActivityFoldersAndPages/Curs%202013-2014/CHomotopy/chomotopy.aspx">Conference on Type Theory, Homotopy Theory and Univalent Foundations</a> (2013) (<a href="http://www.crm.cat/en/Activities/Documents/AbstractsTypeTheory.pdf">talk abstracts pdf</a>).</li> </ul> <p>A writeup finally appeared as:</p> <ul> <li id="FFLL16"><a class="existingWikiWord" href="/nlab/show/Kuen-Bang+Hou">Kuen-Bang Hou</a> (<a class="existingWikiWord" href="/nlab/show/Favonia">Favonia</a>), <a class="existingWikiWord" href="/nlab/show/Eric+Finster">Eric Finster</a>, <a class="existingWikiWord" href="/nlab/show/Dan+Licata">Dan Licata</a>, <a class="existingWikiWord" href="/nlab/show/Peter+LeFanu+Lumsdaine">Peter LeFanu Lumsdaine</a>, <em>A mechanization of the Blakers-Massey connectivity theorem in Homotopy Type Theory</em>, LICS ‘16 (2016) 565–574 [<a href="https://arxiv.org/abs/1605.03227">arXiv.1605.03227</a>, <a href="https://doi.org/10.1145/2933575.2934545">doi:10.1145/2933575.2934545</a>]</li> </ul> <p>Another unwinding to ordinary mathematical language of the above <a href="#Favonia">code</a> was meanwhile given in</p> <ul> <li id="Rezk14"><a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, <em>Proof of the Blakers-Massey theorem</em>, 2014 <a href="https://rezk.web.illinois.edu/freudenthal-and-blakers-massey.pdf">pdf</a>.</li> </ul> <p>prompted by online discussion at</p> <ul> <li id="Schreiber14"> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em>Explaining the point of HoTT on FOM</em>, Google+ thread 2014-10-22 (<a href="https://github.com/DavidMichaelRoberts/Sandbox/blob/master/Schreiber_Gplus_post.md">archived version</a>)</p> <p>(scroll down a fair bit through the list of replies there to see the exchange between <a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a> and <a class="existingWikiWord" href="/nlab/show/Favonia">Favonia</a>).</p> </li> </ul> <p>Further developments along these lines are in</p> <ul> <li id="AnelBiedermanFinsterJoyal17a"> <p><a class="existingWikiWord" href="/nlab/show/Mathieu+Anel">Mathieu Anel</a>, <a class="existingWikiWord" href="/nlab/show/Georg+Biedermann">Georg Biedermann</a>, <a class="existingWikiWord" href="/nlab/show/Eric+Finster">Eric Finster</a>, <a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <em>A Generalized Blakers-Massey Theorem</em>, Journal of Topology <strong>13</strong> 4 (2020) 1521-1553 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/1703.09050">arXiv:1703.09050</a>, <a href="https://doi.org/10.1112/topo.12163">doi:10.1112/topo.12163</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> <blockquote> <p>(generalizing to the case where the <a class="existingWikiWord" href="/nlab/show/%28n-connected%2C+n-truncated%29+factorization+system">(n-connected, n-truncated) factorization system</a> may be replaced by more general <a class="existingWikiWord" href="/nlab/show/modal+homotopy+type+theory">modalities</a>)</p> </blockquote> </li> <li id="AnelBiedermanFinsterJoyal17b"> <p><a class="existingWikiWord" href="/nlab/show/Mathieu+Anel">Mathieu Anel</a>, <a class="existingWikiWord" href="/nlab/show/Georg+Biedermann">Georg Biedermann</a>, <a class="existingWikiWord" href="/nlab/show/Eric+Finster">Eric Finster</a>, <a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <em>Goodwillie’s Calculus of Functors and Higher Topos Theory</em> (<a href="https://arxiv.org/abs/1703.09632">arXiv:1703.09632</a>)</p> </li> </ul> <h3 id="in_shape_theory">In shape theory</h3> <ul> <li>Šime Ungar, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-Connectedness of inverse systems and applications to shape theory, Glasnik Matematički 13 (1978), 371-396 <a href="http://www.irb.hr/korisnici/zskoda/ungarConnwoabsh.pdf">pdf</a></li> </ul> <blockquote> <p>Let (X, A, x) be an n-connected inverse system of CW-pairs such that the restriction (A, x) is m-connected. We prove that there exists an isomorphic inverse system (Y, B, y) having n-connected terms such that the terms of the restriction (B, y) are m-connected. This result is then applied in proving analogues of Hurewicz and Blakers-Massey theorems for homotopy pro-groups and shape groups.</p> </blockquote> </body></html> </div> <div class="revisedby"> <p> Last revised on March 28, 2024 at 22:07:37. 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