2.2 Homology groups and their computation

Homology groups are a key concept in algebraic topology, measuring the "holes" in a topological space. They're defined as quotient modules of cycles and boundaries, capturing essential structural information about the space.

Computing homology groups involves tools like free resolutions and long exact sequences. These techniques allow us to break down complex structures into simpler components, making it easier to calculate and understand the homology of various mathematical objects.

Homology Groups and Their Properties

Definition and Components of Homology Groups

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• Homology group $H_n(C_*)$ defined as the quotient module $\ker d_n / \text{im } d_{n+1}$
• Kernel $\ker d_n$ consists of n-cycles, elements of $C_n$ that are mapped to zero by the boundary map $d_n$
• Image $\text{im } d_{n+1}$ consists of n-boundaries, elements of $C_n$ that are the image of the boundary map $d_{n+1}$ from $C_{n+1}$
• Quotient module $\ker d_n / \text{im } d_{n+1}$ identifies n-cycles that differ by an n-boundary, capturing the "essential" n-dimensional holes in the complex

Invariants of Homology Groups

• Betti numbers $\beta_n$ represent the rank of the free part of the n-th homology group $H_n(C_*)$
  • $\beta_0$ counts the number of connected components
  • $\beta_1$ counts the number of 1-dimensional holes or "loops"
  • $\beta_2$ counts the number of 2-dimensional voids or "cavities"
• Torsion of $H_n(C_*)$ captures the non-free part of the homology group
  • Torsion elements have finite order, i.e., they become zero when multiplied by some non-zero scalar
  • Torsion can be represented by a list of the orders of the torsion subgroups (e.g., $\mathbb{Z}_2, \mathbb{Z}_3$)

Computational Tools for Homology

Free Resolutions

• Free resolution of a module $M$ is an exact sequence of free modules $F_*$ with a map $\varepsilon: F_0 \to M$ such that the sequence $\cdots \to F_2 \to F_1 \to F_0 \xrightarrow{\varepsilon} M \to 0$ is exact
• Free resolutions can be used to compute homology groups by applying the homology functor $H_*$ to the resolution
  • $H_n(M) \cong H_n(F_*)$ for all $n \geq 0$
• Constructing a free resolution involves finding a generating set for the module and relations among the generators

Long Exact Sequences

• Long exact sequence is a tool for computing homology groups of a chain complex from the homology groups of simpler complexes
• Given a short exact sequence of chain complexes $0 \to A_* \to B_* \to C_* \to 0$ there is an associated long exact sequence in homology $\cdots \to H_n(A_*) \to H_n(B_*) \to H_n(C_*) \xrightarrow{\partial} H_{n-1}(A_*) \to \cdots$
• The connecting homomorphism $\partial$ relates the homology groups of the complexes $A_*, B_*,$ and $C_*$
• Long exact sequences can be used to compute unknown homology groups from known ones by exploiting the exactness property
  • Exactness means that the kernel of each map is equal to the image of the previous map