p(e)] denotes the constant path at p(e); this map is a

homotopy equivalence.

Thus, EW's theorem may be viewed as an explanation of the

homotopy equivalenceHis topics include subgroup complexes as geometries for simple groups,

homotopy equivalence, group cohomology and decompositions, spheres in homology and Quillen's conjecture, and local-coefficient homology and representation theory.

We say that f is a

homotopy equivalence if [f] [member of] A(X, Y)/ ~ is an isomorphism in Mor A/~

It is then clear that f restricts, with the help of the cellular approximation theorem, to a rational

homotopy equivalence from the subcomplex

If f : X [right arrow] X is a map and h : X [right arrow] Y is a digital

homotopy equivalence with a homotopy inverse k : Y [right arrow] X, then

The

homotopy equivalence given by Quillen's fiber lemma also carries the action of the symmetric group.

1], but the proof will involve the controlled

homotopy equivalence and the thin h-cobordism theorem.

Note that for every G-space X there is a G-map X x EG [right arrow] X which is a

homotopy equivalence (out of a free G-space).

It has been known that stable Kneser graphs are related to spheres, for example Bjorner and de Longueville (2003) have shown the

homotopy equivalence N([SG.

Observe that a is both a loop map and a rational

homotopy equivalence.

n] (Y) for all n > 0, then the map f is a

homotopy equivalence.