is a homotopy equivalence
. Notice that the pullback of the fibrewise tangent bundle [tau](p x q) restricts to the pullback of the fibrewise tangent bundle of a* [~.Y] [right arrow] X: the fibre at ([~.x], [~.y],[alpha], [beta]) is equal to [[tau].sub.[~.x]] (p) [direct sum] [[tau].sub.[~.y]](q).
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.
It is then clear that f restricts, with the help of the cellular approximation theorem, to a rational homotopy equivalence
from the subcomplex !summation of^K to the indicated subbouquet of spheres.
The lefthand map sends a point e [member of] E to the point (e, [const.sub.p(e)]), where [const.sub.p(e)] denotes the constant path at p(e); this map is a homotopy equivalence
. The right-hand map sends a point (e, [phi]) [member of] mps(p) to the point [phi] (1) [member of] B; this map is a fibration.
The homotopy equivalence
given by Quillen's fiber lemma also carries the action of the symmetric group.
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
More is true: There is a natural homotopy equivalence
[absolute value of Hom(G, H)] [equivalent] [absolute value of Hom(1, [G, H])] induced by a poset map which preserves atoms and with a homotopy inverse of the same kind.
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).
The constructions [L.sub.r], [[bar.L].sup.r] induce classifications (for flows) of the following type: Two flows X, Y are said to be [[bar.L].sup.r]-equivalent if there is a flow morphism f: X [right arrow] Y such that [[bar.L].sup.r] (f) is a homotopy equivalence
. This will determine equivalence classes of flows having, for instance, the same number of critical points or the 'same type' of periodic trajectories.
By Lemma 4.4 there is a homotopy equivalence
e: [SIGMA]A' [disjunction] [S.sup.17] [right arrow] [SIGMA]B/[S.sup.6] where the restriction of e to [SIGMA]A' is [SIGMA]i'.