Hatcher-Algebraic-Topology · Problem 2.2.16

Homology

Mathematics — Graph Theory · simplicial-homology-computation

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\bigl((\Delta^n)^k\bigr)

equal to

0

for

i < k

, and free of rank

\binom{n}{k+1}

for

i = k

Intuitive explanation

The rank of $\Delta_i(\Delta^n)$ counts the number of $i$ -dimensional faces—which is $\binom{n+1}{i+1}$ by choosing $i+1$ vertices from $n+1$ total. The boundary map relates consecutive chain groups, and Pascal's triangle emerges because the rank of cycles minus boundaries follows the binomial coefficient recurrence. For the $k$ -skeleton, dimensions below $k$ have no $k$ -cells to create new cycles, so homology vanishes; at dimension $k$ , the cycles that aren't boundaries form a free group whose rank is determined by the deficit between $k$ -chains and their images under the boundary map.

Intuitive explanation

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