Algebraic Topology · Problem 1

Cohomology — Allen Hatcher

Mathematics — Topology · cup-product-surfaces

Problem

Assuming as known the cup product structure on the torus

S^1 \times S^1

, compute the cup product structure in

H^*(M_g)

for

M_g

the closed orientable surface of genus

g

by using the quotient map from

M_g

to a wedge sum of

g

tori, shown below.