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 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.

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'.