nLab
higher homotopy van Kampen theorem

This theorem for filtered spaces involves the fundamental crossed complex $\Pi X_{*}$ of a filtered space $X_{*}$, and the notion of connected filtered space.

Statement

Suppose $X_{*}$ is a filtered space and $X$ is the union of the interiors of sets $U^i$, $i\in I$. Let $U_{*}^i$ be the filtered space given by the intersections $U^i\cap X_n$ for $n\ge 0$. If $d=(i,j)\in I^2$ we write $U^d$ for $U^i\cap U^j$. We then have a coequaliser diagram of filtered spaces

$$\underset{d\in I^2}{⨆}{U}_{*}^d{\rightrightarrows }_b^a\underset{i\in I}{⨆}{U}_{*}^i{\to }^c{X}_{*}.$$\bigsqcup_{d \in I^2} U^d_* \rightrightarrows ^a_b \bigsqcup _{i \in I} U^i_* \to ^c X_*.

Higher Homotopy van Kampen Theorem

If the filtered spaces $U_{*}^f$ are connected filtered spaces for all finite intersections $U_{*}^f$ of the filtered spaces $U_{*}^i$, then

1. (Conn) The filtered space $X_{*}$ is connected; and
2. (Iso) The fundamental crossed complex functor $\Pi$ takes the above coequaliser diagram of filtered spaces to a coequaliser diagram of crossed complexes.

Remarks

• Note that because $\Pi$ uses groupoids, it obviously takes disjoint unions $⨆$ of filtered spaces into disjoint unions (= coproducts) $⨆$ of crossed complexes.

• The proof of the theorem is not direct but goes via the fundamental cubical omega-groupoid? with connections of the filtered spaces, as that context allows the notions of algebraic inverse to subdivision and of commutative cube. However the proof is a direct generalisation of a proof for the van Kampen theorem for the fundamental groupoid.

• Applications of this theorem include many basic facts in algebraic topology, such as the Relative Hurewicz Theorem, the Brouwer degree theorem, and new nonabelian results on 2nd relative homotopy groups, not of course obtainable by the traditional wholly abelian methods. No use is made of singular homology theory or of simplicial approximation.

Sample application

Here is one application in dimension 2 not easily obtainable by traditional algebraic topology. Let $0\to P\to Q\to R\to 0$ be an exact sequence of abelian groups. Let $X$ be the mapping cone of the induced map $K(P\mathrm{,1})\to K(Q\mathrm{,1})$ of Eilenberg-Mac Lane spaces. Then a crossed module representing the homotopy 2-type of $X$ is $\mu :C\to Q$ where $C$ is abelian and is the direct sum ${\oplus }_{r\in R}{P}^r$ of copies of $P$ one for each $r\in R$ and the action of $Q$ is via $R$ and permutes the copies by $(p,r{)}^s=(p,r+s)$. Similar examples for $P,Q,R$ nonabelian are do-able, more complicated, and certainly not obtainable by traditional methods.