Algebraic Topology · Problem 2.2.16

Homology — Allen Hatcher

Mathematics — Topology · simplicial-homology-computations

Problem

Let

\Delta^n = [v_0, \cdots, v_n]

have its natural

\Delta

-complex structure with

k

-simplices

[v_{i_0}, \cdots, v_{i_k}]

for

i_0 < \cdots < i_k

. Compute the ranks of the simplicial (or cellular) chain groups

\Delta_i(\Delta^n)

and the subgroups of cycles and boundaries. [Hint: Pascal's triangle.] Apply this to show that the

k

-skeleton of

\Delta^n

has homology groups

\widetilde{H}_i\left((\Delta^n)^k\right)

equal to 0 for

i < k

, and free of rank

\binom{n}{k+1}

for

i = k