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 equivalence
His 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
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