Algebraic Topology · Problem 2.1.8

Homology — Allen Hatcher

Mathematics — Topology · simplicial-homology-computations

Problem

Construct a 3-dimensional

\Delta

-complex

X

from

n

tetrahedra

T_1, \cdots, T_n

by the following two steps. First arrange the tetrahedra in a cyclic pattern as in the figure, so that each

T_i

shares a common vertical face with its two neighbors

T_{i-1}

and

T_{i+1}

, subscripts being taken mod

n

. Then identify the bottom face of

T_i

with the top face of

T_{i+1}

for each

i

. Show the simplicial homology groups of

X

in dimensions 0, 1, 2, 3 are

\mathbb{Z}

\mathbb{Z}_n

0

\mathbb{Z}

, respectively. [The space

X

is an example of a lens space; see Example 2.43 for the general case.]