Homology of torus

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

$$(S^1)^n \cong S^1 \times S^1 \times \dots S^1$$

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 $$k^{th}$$ homology group $$H_k(T^n)$$ is a free abelian group $$\mathbb{Z}^{\binom{n}{k}}$$ 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 $$0 \le k \le n$$ and zero for $$k > n$$.

Reduced version over integers
The $$k^{th}$$ 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 $$k^{th}$$ homology group $$H_k(T^n;M)$$ is a direct sum $$M^{\binom{n}{k}}$$ 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.

Homology groups in tabular form
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 $$k$$ for which we are computing $$H_k(T^n)$$. If a cell value reads 2, for instance, that means that the corresponding homology group with coefficients in the integers is $$\mathbb{Z}^2 = \mathbb{Z} \oplus \mathbb{Z}$$ and the corresponding homology group with coefficients in a module $$M^2$$. Note that the cell values for $$k > n$$ are omitted, because all these values are zero:

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