Homology of torus

This article describes the value (and the process used to compute it) of some homotopy invariant(s) for a topological space or family of topological spaces. The invariant is homology and the topological space/family is torus
Get more specific information about torus | Get more computations of homology

Statement

We denote by $T^{n}$ the $k$ -dimensional torus, which is the topological space:

$(S^{1})^{n} ≅ S^{1} \times S^{1} \times \dots S^{1}$

i.e., the product of $S^{1}$ , the circle, with itself $n$ times. It is equipped with the product topology. If we think of $S^{1}$ as a group, $T^{n}$ gets a group structure too as the external direct product.

Unreduced version over integers

The $k^{t h}$ homology group $H_{k} (T^{n})$ is a free abelian group $Z^{(\binom{n}{k})}$ of rank $(\binom{n}{k})$ , where $(\binom{n}{k})$ denotes the binomial coefficient, or the number of subsets of size $k$ in a set of size $n$ . In particular, $(\binom{n}{k})$ is positive for $k \in {0, 1, 2, \dots, n}$ and zero for other $k$ . Thus, $H_{k} (T^{n})$ is nontrivial for $0 \leq k \leq n$ and zero for $k > n$ .

Below are given the ranks of homology groups for small values of $n$ and $k$ . Each row corresponds to a value of $n$ and each column corresponds to a value of $k$ . If a cell value reads 2, for instance, that means that the corresponding homology group is $Z^{2} = Z \oplus Z$ :

$n, k$	0	1	2	3	4
0	1	0	0	0	0
1	1	1	0	0	0
2	1	2	1	0	0
3	1	3	3	1	0
4	1	4	6	4	1

Reduced version over integers

The $k^{t h}$ reduced homology group ${\tilde{H}}_{k} (T^{n})$ is a free abelian group of rank $(\binom{n}{k})$ for $k > 0$ and is trivial for $k = 0$ . In particular, it is a nontrivial group for $k \in {1, 2, \dots, n}$ and is zero for other $k$ .

Unreduced version over an abelian group

The $k^{t h}$ homology group $H_{k} (T^{n}; M)$ is a direct sum Failed to parse (syntax error): {\displaystyle M}^{\binom{n}{k}}} of rank $(\binom{n}{k})$ , where $(\binom{n}{k})$ denotes the binomial coefficient, or the number of subsets of size $k$ in a set of size $n$ . The behavior is qualitatively the same as over the integers. Note that this result is the same regardless of whether we think of the homology with coefficients in $M$ as an abelian group or as a module over some other commutative unital ring.

Reduced version over an abelian group

This is the same as the unreduced version, except that the zeroth homology group is zero.

Related invariants

These are all invariants that can be computed in terms of the homology groups.

Invariant	General description	Description of value for torus
Betti numbers	The $k^{t h}$ Betti number $b_{k}$ is the rank of the $k^{t h}$ homology group.	$b_{k} = (\binom{n}{k})$
Poincare polynomial	Generating polynomial for Betti numbers	$(1 + x)^{n}$
Euler characteristic	$\sum_{k = 0}^{\infty} (- 1)^{k} b_{k}$	0 -- can also be seen from the fact that we have a group.

Automorphisms, endomorphisms, and homomorphisms