The n-th homology group of a chain complex C_* is the group
H_n = \ker(\partial_n)/\text{Im}(\partial_{n+1})
For an abstract simplicial complex \Sigma, its homology with coefficients in a field \mathbb k is the homology
H_*(\Sigma, k) = H_*(C_*(\Sigma, \mathbb k))
Given \varepsilon_1 < \varepsilon_2 we define persistent homology
H^{\varepsilon_1 \to \varepsilon_2}_m(Z) = \text{Im}\{i_* : H_m(\text{VR}(Z, \varepsilon_1)) \to H_m(\text{VR}(Z, \varepsilon_2))\}
A chain map f_*: C_* \to D_* is graded map of degree 0, C_n \to D_n such that f_{n-1} \circ \partial_n = \partial_n \circ f_n
A chain map f_* induces a linear transformation H_n(f_*) : H_n(C_*) \to H_n(D_*) such that
Homology is a functor from chain complexes to graded abelian groups.
Let I_* be the chain complex of a unit interval (1-simplex).
A chain homotopy from a chain map f_*: C_* \to D_* to a chain map g_*: C_* \to D_* is a graded map of degree 1 s_*: C_* \to D_* such that \partial_*s_* + s_*\partial_* = g_* - f_*
Every continuous map f: D^n \to D^n has a fixed point.
Where D^n = \{x \in \mathbb R^n : ||x|| \le 1\}
Given a diagram of linear transformations
U \stackrel{L}{\longrightarrow} V \stackrel{M}{\longrightarrow} W
the sequence of linear transformations is exact if \text{im}(L) = \ker(M) and M \circ L = 0.
It is exact iff \dim(V) = \text{rank}(L) + \text{rank}(M).
Given a simplicial complex X which is the union of two subcomplexes Y and Z (note that Y \cap Z is also a subcomplex of X). The long exact sequence is
\cdots \longrightarrow H_i(Y \cap Z) \stackrel{\alpha_i}{\longrightarrow} H_i(Y) \oplus H_i(Z) \longrightarrow H_i(X) \stackrel{\delta}{\longrightarrow} H_{i-1}(Y \cap Z) \stackrel{\alpha_{i_1}}{\longrightarrow} \cdots
With \alpha_i(\xi) = (H_i(i_0)(\xi), -H_i(i_1)(\xi)) where i_0 : Y \cap Z \hookrightarrow Y and i_1 : Y \cap Z \hookrightarrow Z.
The dimension of H_i(X) is
\dim(H_i(Y)) + \dim(H_i(Z)) + \dim(H_{i-1}(Y \cap Z)) - \text{rank}(\alpha_i) - \text{rank}(\alpha_{i-1})
Denoted by \tilde H_n
H_n(X) = \begin{cases} \tilde H_n(X) & \text{ for } n > 0 \\ \tilde H_0(X) \oplus \mathbb Z & \text{ for } n = 0 \\ \end{cases}
Given a simplicial complex X and its subcomplex Y we have we have chain complexes C_*(X) and C_*(Y) which form a quotient vector spaces C_i(X)/C_i(Y) denoted by C_i(X, Y). This induces boundary maps \partial: C_i(X, Y) \to C_{i-1}(X, Y). With those we can compute homology H_i(X, Y) called the relative homology of X with respect to Y. This creates a long exact sequence:
\cdots \longrightarrow H_{i+1}(X, Y) \longrightarrow H_i(Y) \longrightarrow H_i(X) \longrightarrow H_i(X, Y)\longrightarrow H_{i-1}(Y) \longrightarrow \cdots
\tilde H_*(X / A) \simeq H_*(X, A)
Let Z \subseteq Y \subseteq X be inclusions of subcomplexes. The natural chain map induces an isomorphism H_*(X, Y) \simeq H_*(X - Z, Y - Z).