Hatcher-Algebraic-Topology · Problem 2.2.29

Homology

Mathematics — Graph Theory · mayer-vietoris-sequence

Problem

The surface

M_g

of genus

g

, embedded in

\mathbb{R}^3

in the standard way, bounds a compact region

R

. Two copies of

R

, glued together by the identity map between their boundary surfaces

M_g

, form a closed 3-manifold

X

. Compute the homology groups of

X

via the Mayer--Vietoris sequence for this decomposition of

X

into two copies of

R

. Also compute the relative groups

H_i(R, M_g)