nLab
filtered topological space

Contents

Definition

A filtered topological space X *X_* is a filtered object in Top, hence

  1. a topological space X=X X=X_\infty

  2. equipped with a sequence of subspaces

    X *:=X 0X 1X nX .X_*:= \quad X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n \subseteq \cdots \subseteq X_\infty.

A filtered space X *X_* is called a connected filtered space if it satisfies the following condition:

(ϕ) 0(\phi)_0: The function π 0X 0π 0X r\pi_0X_0 \to \pi_0 X_r induced by inclusion is surjective for all r0r \geq 0; and, for all i1i \geq 1, (ϕ i):π i(X r,X i,v)=0(\phi_i): \pi_i(X_r,X_i,v)=0 for all r>ir \gt i and vX 0 v \in X_0.

There are two other forms of this condition which are useful under different circumstances.

Examples

  1. A CW-complex XX with its filtration by skeleta X nX^n.

  2. The free topological monoid FXF X on a space XX filtered by the length of words. Given a based space (X,x)(X,x), there is also a reduced version by taking FXF X and identifying xx with the identity of FXF X. This latter filtered space is known as the James construction J(X,x)J(X,x), after Ioan James.

    The James construction J(X,x)J(X,x) can be constructed homotopy-theoretically, following ideas of Brunerie. Recall that, for KK a finite simplicial complex, for (X,A)(X,A) a pair of spaces, its polyhedral product (X,A) K(X,A)^K is defined as the union σS(K)(X,A) σ\bigcup_{\sigma\in S(K)}(X,A)^\sigma as a subspace of the Cartesian product X V(K)X^{V(K)}. Here, for σS(K)\sigma\in S(K) a simplex of KK, the subspace (X,A) σ(X,A)^\sigma consist of those xX V(K)x\in X^{V(K)} such that, for each vertex vv in the complement of σ\sigma, the coordinate projection proj vx\proj_v x lies in AA. Equivalently, the polyhedral product (X,A) K(X,A)^K can be considered as a homotopy colimit of these (X,A) σ(X,A)^\sigma over the poset S(K)S(K) of simplexes σ\sigma of KK, where the maps are the respective inclusions.

    Definition

    For XX a space equipped with a basepoint xx, define a filtered space fil fil_\bullet as follows. Set fil 0\fil_0 as {x}\{x\}. For k1k\ge 1, require that the following square is homotopy pushout:

    (X,x) Δ[k1] inc X k fil k1 p k j k fil k \array{ &&&& (X,x)^{\partial \Delta[k-1]} &&&& \\ & && inc \swarrow & & \searrow && \\ && X^k &&&& fil_{k-1} \\ & && {}_{p_k}\searrow & & \swarrow_{j_k} && \\ &&&& fil_k &&&& }

    where the unlabeled arrow is the homotopy colimit of a morphism of diagrams over S(Δ[k1])S(\partial \Delta[k-1]) given by the maps

    (X,x) σX dim(σ)+1p dim(σ)+1fil dim(σ)+1(X,x)^\sigma \xrightarrow{\sim} X^{\dim(\sigma)+1} \xrightarrow{p_{\dim(\sigma)+1}} fil_{\dim(\sigma)+1}

    for each simplex σ\sigma of the boundary simplicial complex Δ[k1]\partial \Delta[k-1] of the standard (k1)(k-1)-simplex.

    Proposition

    For (X,x)(X,x) a pointed space, if (X,x)(X,x) is path-connected, then fil ΩΣXfil_\infty \simeq \Omega\Sigma X.

  3. A similar example to the last using free groups instead of free monoids.

  4. A similar example to the last using free groupoids on topological graphs.

  5. A similar example to the last using the universal topological groupoid U σ(G)U_\sigma(G) induced from a topological groupoid GG by a continuous function f:Ob(G)Yf: Ob(G) \to Y to a space YY.

Examples of connected filtered spaces are:

  1. The skeletal filtration of a CW-complex.

  2. The word length filtration of the James construction for a space with base point such that {x}X\{x\} \to X is a closed cofibration.

  3. The filtration (BC) *(B C)_* of the classifying space of a crossed complex, filtered using skeleta of CC.

This condition occurs in the higher homotopy van Kampen theorem for crossed complexes.

Properties

We thus see that filtered spaces arise from many geometric and algebraic situations, and see also stratified spaces). It is therefore interesting that one can define strict higher homotopy groupoids for filtered spaces more easily than for spaces themselves.

Note also that it is standard to be able to replace, using mapping cylinders, a sequence of maps Y nY n+1Y_n \to Y_{n+1} by a sequence of inclusions.

References

Guillaume Brunerie gave a talk entitled “The James Construction and π 4(S 3)\pi_4(S^3)” at the Institute of Advanced Studies on March 27, 2013. In this talk, he described James construction in homotopy type theory.

Revised on July 22, 2014 17:01:52 by David Roberts (129.127.211.231)