Hatcher-Algebraic-Topology · Problem 2.2.33

Homology

Mathematics — Graph Theory · mayer-vietoris-sequence

Problem

Suppose the space

X

is the union of open sets

A_1, \cdots, A_n

such that each intersection

A_{i_1} \cap \cdots \cap A_{i_k}

is either empty or has trivial reduced homology groups. Show that

\widetilde{H}_i(X) = 0

for

i \geq n-1

, and give an example showing this inequality is best possible, for each

n