The theory of Homology groups of Chain complexes derived from Topological spaces, often generalizing concepts of cycles and boundaries. These groups are algebraic quantities which are Topological invariants. Video intro

Basically the fundamental homology group for topological space $X$ is $H_1(X) = \text{1-cycles of }X$ / $\text{1-boundaries of } X$.

hatcher

A 1-cycle is basically a local superposition of lines glued end to end, such that the beginning of the first line is the same point as the ending of the last line

A 1-boundary is basically an equivalence relation saying that two 1-cycles are equivalent when they bound a surface.

higher homology groups are defined with higher-dimensional analogs.

Delta-complexes and Simplicial homology

Singular homology

At the end of this section, after some theory has been developed, we will show that simplicial and singular homology groups coincide for ∆ complexes.

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 Chain map, 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.

Singular homology is the same as simplicial homology when a space admits a $\Delta$-complex . The homomorphisms $H_n^\Delta (X, A) \to H_n (X, A)$ induced by the maps $\Delta_n(X,A) \to C_n(X,A)$ are isomorphisms for all n and all ∆ complex pairs (X, A) .

Reduced homology

Reduced homology group. Makes the homology of the point be trivial.

Reduced homology theory (axiomatized)

Homology group of a Wedge sum of spaces is the Direct sum of homology groups

Theorem that shows that two spaces in two dimensions are not homeomorphic.

Relative homology

Exact sequence –> Homology long exact sequence (very useful to find relations between homologies of spaces and subspaces, using Homological algebra)

Excision theorem

Cellular homology

Homology defined on CW complexes

Bordism and Cobordism

Relation between fundamental group and homology – Proposition: there is a Group homomorphism from the Fundamental group to the 1st Homology group. More precise relation

Brouwer fixed-point theorem

Cohomology

Homological algebra

Homology of product of spaces. Uses Short exact sequence