The singular homology group of a simplicial complex is the group of all continuous maps from the simplicial complex to the integers.

Singular homology is a generalization of simplicial homology in the sense that every simplicial complex can be embedded in a singular complex, and the singular homology groups of the simplicial complex are isomorphic to the singular homology groups of the singular complex.

The homology group of a sphere is $\mathbb{Z}$, which means that it is isomorphic to the group of integers.

The homology group of a torus is $\mathbb{Z}^2$, which means that it is isomorphic to the group of pairs of integers.

The homology group of a Klein bottle is $\mathbb{Z}_2^2$, which means that it is isomorphic to the group of pairs of integers modulo 2.

The homology group of a projective plane is $\mathbb{Z}_2$, which means that it is isomorphic to the group of integers modulo 2.

The homology group of a Mobius strip is $\mathbb{Z}$, which means that it is isomorphic to the group of integers.

The homology group of a disk is $\mathbb{Z}$, which means that it is isomorphic to the group of integers.

The homology group of a cylinder is $\mathbb{Z}$, which means that it is isomorphic to the group of integers.

The homology group of a cone is $\mathbb{Z}$, which means that it is isomorphic to the group of integers.

The homology group of a sphere with $n$ handles is $\mathbb{Z}^n$, which means that it is isomorphic to the group of $n$-tuples of integers.

The homology group of a torus with $n$ holes is $\mathbb{Z}^n$, which means that it is isomorphic to the group of $n$-tuples of integers.

The homology group of a Klein bottle with $n$ handles is $\mathbb{Z}_2^n$, which means that it is isomorphic to the group of $n$-tuples of integers modulo 2.

The homology group of a projective plane with $n$ holes is $\mathbb{Z}_2^n$, which means that it is isomorphic to the group of $n$-tuples of integers modulo 2.

The homology group of a Mobius strip with $n$ twists is $\mathbb{Z}$, which means that it is isomorphic to the group of integers.