proving that an inclusion map from a subcomplex is a homotopy equivalence

Question

This is a pretty basic question but I have been stuck on it for a while.

Given an abstract simplicial complex X and a subcomplex A, why does * suffice to show that the map |A|->|X| induced by inclusion is a homotopy equivalence:

• Let g: (|K|,|L|) -> (|X|,|A|) be a continuous map, where K is a finite simplicial complex and L a subcomplex of K. Any such g is homotopic rel |L| to a map sending |K| into |A|.

Here |.| denotes the geometric realization.

I'm trying to understand the very first step of the proof of Proposition 2.2 of J-C. Hausmann's paper "On the Vietoris-Rips complexes and a cohomology theory for metric spaces".

Answer 1

By taking K = a simplex and L = its boundary you can show that |A| -> |X| is an isomorphism on all homotopy groups (do surjectivity and injectivity separately). Then apply Whitehead's theorem.

Answer 2

You can then apply this to id: (|X|,|A|) -> (|X|,|A|) to get a homotopy, rel A, from the identity of X to a retraction r: X -> A. This shows that A is a strong deformation retract of X and is in particular is a homotopy equivalence.

EDIT: OK. I missed the finiteness hypothesis. If you don't assume the Whitehead theorem (in fact, you usually use something like this to prove the Whitehead theorem), then you need to do the following.

Let X⁽ⁿ⁾ be the union of A with the n-skeleton of X. Suppose we have already constructed a self-homotopy of |X| rel |A| moving |X^(n-1)| into |A|. Call the resulting map f_n-1 from (|X|,|X^(n-1)|) to (|X|,|A|).

You can then, for each n-simplex of X⁽ⁿ⁾ not in A, use the given property (*) to find a homotopy from this simplex, rel its boundary, from the map f_n-1 to a new map moving the simplex into |A|. Putting these together gives you a homotopy rel X^(n-1) from f_n-1 to a new map g that takes |X⁽ⁿ⁾| into |A|.

The inclusion |X⁽ⁿ⁾| -> |X| is a relative CW-inclusion, and so it has the homotopy extension property, and you can extend your given homotopy to a self-homotopy of |X| rel |A| from f_n-1 to a new map f_n that maps |X⁽ⁿ⁾| into the boundary. Then induct.

You get a sequence f_i of functions and homotopies H_i from one to the next. If you glue these together by applying the homotopy H_i on a time inteval of length 1/2ⁱ, you get a finite time homotopy H of |X| rel |A| from the identity to a map sending |X| into |A|. That's your strong deformation retract.