Definition. For a Topological space $X$, we consider the set of all Singular simplexes, the corresponding Chain group, the Boundary maps (vid) between them, giving rise to a Chain complex (called Singular chain complex). Then singular homology is defined as the Singular homology groups, which are the quotient of the kernel of the outgoing map by the image of the incoming map (that is the Homology groups).

Note that the set of singular simplices (continuous functions from simplex to space) is most often uncountably Infinite, and so the Chain groups are also infinite. However, the quotient groups defining the singular homology turn out to be finite. (see here).

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Intuition

Homology is functorial

If we have a Continuous function between topological spaces, this gives us a Group homomorphism between the corresponding homology groups. This will mean that Homology is a Topological invariant, and so can be used to study if two spaces are homeomorphic.

Formal definition of induced Chain map $f_\#$, defined as the pushforward of singular simplex, and extending linearly, and it is easy to check directly that it commutes with boundary map (by using its definition), so it is indeed a chain map.

This chain map induces a map on homology (see below also), defined by lifting f_* to the homology classes, and it is called the pushforward by $f$.

Functoriality under composition: vid

Homotopy invariance

Video1/Video2 Homotopy equivalent spaces have isomorphic Homology groups. See also Hatcher. We define a Functor by finding an induced function $f_\#$ between Chain groups which is a Group homomorphism, which itself induces a function $f_*$ between Homology groups by lifting $f_\#$ to the Cosets defining the homology groups, and showing it's well defined because it maps cycles to cycles and boundaries to boundaries. Finally, there is a theorem that shows that homotopic maps lead to equal $f_*$, from which one can conclude (after showing Functoriality) that homotopy equivalences induce group isomorphisms between the homology groups.

Proposition. If you decompose the space into Path-connected components, then the homology group of the space is the Direct sum of the homology groups of the components.

Proposition If $X$ is non-emtpy and path connected, then the zeroth homology group $H_0(X) = \mathbb{Z}$ (the Ring of Integers). Hence for any space $X$, $H_0(X) = \bigoplus \mathbb{Z}$ (direct sum of $\mathbb{Z}$s, one for each path-connected component). That is, for a path-connected space, all points are homologic (as they are the boundary of a path), and so the homology group is the free abelian group generated by a single generator (the homology class of any point), which is isomorphic to $\mathbb{Z}$.

Prop 2.8. If $X$ is a point, then $H_n(X)=0$ for $n>0$ and $H_0(X) \cong \mathbb{Z}$.

It is often very convenient to have a slightly modified version of homology for which a point has trivial homology groups in all dimensions, including zero. This is e n (X) to be the homology groups done by defining the Reduced homology groups

Cohomology can be computed using differential forms!.