Algebraic Topology · Problem 2

Cohomology — Allen Hatcher

Mathematics — Topology · cup-product-suspension

Problem

Using the cup product

H^k(X,A;R) \times H^\ell(X,B;R) \to H^{k+\ell}(X, A \cup B; R)

, show that if

X

is the union of contractible open subsets

A

and

B

, then all cup products of positive-dimensional classes in

H^*(X;R)

are zero. This applies in particular if

X

is a suspension. Generalize to the situation that

X

is the union of

n

contractible open subsets, to show that all

n

-fold cup products of positive-dimensional classes are zero.