20 Jack Palmer Sanders
Suppose X and Y are pointed spaces each having the homotopy type of
a CW complex. Then X A Y has the homotopy type of a CW complex. If
(X,A) has the homotopy type of a CW pair, then X/A has the homotopy type
of a CW complex. Given f: X - Y, one can show that Cy(f) and C(f)
have the homotopy type of CW complexes and that (Cy(f),X A 0) has the
homotopy type of a CW pair. We need the following well-known results.
1.14. Proposition. Let B be a closed subspace of a pointed space Y
such that (Y,B) has the homotopy type of a CW pair. Let (Y,B) be
n-connected, n 2, and let B be m-connected, m 1. Then the projection
map TT: (Y,B) - (Y/B,*) induces TT. : T T (Y,B) - T T (Y/B,*), an isomorphism
* q q
for q _ m+n and an epimorphism for q = m+n+1.**
1.15. Proposition. Let X be (p-1)-connected, and let Y be
(q-1)-connected, where X and Y are pointed spaces having the homotopy
type of CW complexes. Then X A Y is (p+q-1)-connected.**
1.16. Proposition. Let f: X - Y be a map of pointed spaces, where
X is n-connected and Y is m-connected. Then the inclusion
i: (Y,*) - (Cy(f),X A 0) induces an isomorphism
i*: Tq(Y,*) T - Tq(Cy(f),X T A 0) for q n. If m n, then (Cy(f),X A 0)
* - -
is m-connected.
Proof. This follows from the homotopy sequence of the pair
(Cy(f),X A 0) and the fact that the inclusion Y c Cy(f) is a homotopy
equivalence.**
Let Z denote the integers, and let 4Z = {4n|n e Z}.
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