1.2 Simplicial homology

Simplicial homology bridges topology and algebra, using discrete structures to study continuous spaces. It breaks down complex shapes into simpler pieces, allowing us to analyze their topological features using algebraic tools.

At its core, simplicial homology uses simplicial complexes to represent spaces and chain complexes to capture their structure. By examining cycles and boundaries, we can identify "holes" in the space and compute homology groups, revealing key topological properties.

Simplicial complexes

• Simplicial complexes are combinatorial objects used to represent topological spaces in a discrete manner
• They provide a way to study the topological properties of spaces using algebraic techniques
• Simplicial complexes form the foundation for simplicial homology, a key tool in cohomology theory

Definition of simplicial complexes

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• A simplicial complex is a collection of simplices (points, edges, triangles, tetrahedra, etc.) that fit together in a specific way
• The simplices must satisfy certain conditions:
  • The intersection of any two simplices is either empty or a common face of both simplices
  • Every face of a simplex in the complex is also in the complex
• Formally, a simplicial complex $K$ is a set of simplices such that if $\sigma \in K$ and $\tau \subseteq \sigma$, then $\tau \in K$

Examples of simplicial complexes

• A graph is a 1-dimensional simplicial complex consisting of vertices (0-simplices) and edges (1-simplices)
• A triangulation of a surface, such as a sphere or a torus, is a 2-dimensional simplicial complex
• The boundary of a tetrahedron is a 2-dimensional simplicial complex consisting of four triangular faces

Geometric realization of simplicial complexes

• The geometric realization of a simplicial complex is a topological space obtained by gluing together the simplices according to their combinatorial structure
• Each simplex is realized as a geometric object (point, line segment, triangle, etc.) in a Euclidean space
• The geometric realization provides a way to visualize and study the topological properties of the simplicial complex

Oriented simplicial complexes

• Oriented simplicial complexes are simplicial complexes with an additional structure that assigns an orientation to each simplex
• The orientation is crucial for defining the boundary operator and studying homology

Ordering of vertices

• To define an orientation on a simplicial complex, we first order the vertices of each simplex
• The ordering of vertices determines the orientation of the simplex
• For example, in a 2-simplex (triangle), the ordering $(v_0, v_1, v_2)$ gives a positive orientation, while $(v_0, v_2, v_1)$ gives a negative orientation

Oriented simplices

• An oriented simplex is a simplex with a chosen ordering of its vertices
• We denote an oriented simplex by $[v_0, v_1, \ldots, v_n]$, where $v_0, v_1, \ldots, v_n$ are the vertices in the chosen order
• Changing the order of vertices changes the orientation of the simplex by a sign $(-1)^{\text{number of transpositions}}$

Boundary operator on oriented simplices

• The boundary operator $\partial$ on an oriented $n$-simplex $[v_0, v_1, \ldots, v_n]$ is defined as: $\partial[v_0, v_1, \ldots, v_n] = \sum_{i=0}^{n} (-1)^i [v_0, \ldots, \hat{v_i}, \ldots, v_n]$ where $\hat{v_i}$ means omitting the vertex $v_i$
• The boundary of an oriented simplex is a linear combination of its oriented faces with alternating signs
• The boundary operator satisfies the property $\partial \circ \partial = 0$, which is crucial for defining homology

Chain complexes

• Chain complexes are algebraic objects that capture the combinatorial structure of simplicial complexes
• They provide a way to study the homology of a space using linear algebra

Definition of chain complexes

• A chain complex $C_*$ is a sequence of abelian groups (or vector spaces) $C_n$ connected by boundary operators $\partial_n: C_n \to C_{n-1}$
• The boundary operators satisfy the condition $\partial_{n-1} \circ \partial_n = 0$ for all $n$
• In the context of simplicial homology, $C_n$ is the free abelian group generated by the oriented $n$-simplices of a simplicial complex

Boundary operator on chain complexes

• The boundary operator $\partial_n: C_n \to C_{n-1}$ is defined by extending the boundary operator on oriented simplices linearly
• For a chain $c = \sum_i a_i \sigma_i \in C_n$, where $a_i$ are coefficients and $\sigma_i$ are oriented $n$-simplices, the boundary is: $\partial_n(c) = \sum_i a_i \partial(\sigma_i)$
• The boundary operator on chain complexes satisfies $\partial_{n-1} \circ \partial_n = 0$, which follows from the property of the boundary operator on oriented simplices

Cycles and boundaries

• A cycle in a chain complex $C_*$ is a chain $c \in C_n$ such that $\partial_n(c) = 0$
• The set of $n$-cycles is denoted by $Z_n = \ker(\partial_n)$
• A boundary in a chain complex $C_*$ is a chain $c \in C_n$ such that there exists a chain $d \in C_{n+1}$ with $c = \partial_{n+1}(d)$
• The set of $n$-boundaries is denoted by $B_n = \operatorname{im}(\partial_{n+1})$
• The property $\partial \circ \partial = 0$ implies that every boundary is a cycle, i.e., $B_n \subseteq Z_n$

Homology groups

• Homology groups are algebraic objects that measure the "holes" in a topological space
• They are defined using the cycles and boundaries of a chain complex

Definition of homology groups

• The $n$-th homology group $H_n$ of a chain complex $C_*$ is defined as the quotient group: $H_n = Z_n / B_n = \ker(\partial_n) / \operatorname{im}(\partial_{n+1})$
• Intuitively, homology groups capture the cycles that are not boundaries, i.e., the "holes" that cannot be filled by higher-dimensional chains
• The rank of the homology group $H_n$ is called the $n$-th Betti number and is denoted by $\beta_n$

Computations of homology groups

• To compute the homology groups of a simplicial complex, we first construct the corresponding chain complex
• Then, we compute the kernel and image of the boundary operators to find the cycles and boundaries
• Finally, we take the quotient of the cycles by the boundaries to obtain the homology groups
• In practice, this can be done using linear algebra techniques, such as Smith normal form

Examples of homology group calculations

• For a contractible space, such as a simplex or a ball, all homology groups are trivial (i.e., $H_n = 0$ for all $n$)
• For a circle $S^1$, the homology groups are $H_0 = \mathbb{Z}$, $H_1 = \mathbb{Z}$, and $H_n = 0$ for $n \geq 2$
• For a torus $T^2$, the homology groups are $H_0 = \mathbb{Z}$, $H_1 = \mathbb{Z} \oplus \mathbb{Z}$, $H_2 = \mathbb{Z}$, and $H_n = 0$ for $n \geq 3$

Simplicial maps

• Simplicial maps are morphisms between simplicial complexes that preserve the combinatorial structure
• They induce homomorphisms on the corresponding chain complexes and homology groups

Definition of simplicial maps

• A simplicial map $f: K \to L$ between simplicial complexes $K$ and $L$ is a function that maps vertices of $K$ to vertices of $L$ such that:
  • If $\{v_0, \ldots, v_n\}$ is a simplex in $K$, then $\{f(v_0), \ldots, f(v_n)\}$ is a simplex in $L$
• Simplicial maps preserve the structure of simplicial complexes and can be thought of as "continuous" maps in the combinatorial setting

Induced homomorphisms on chain complexes

• A simplicial map $f: K \to L$ induces a homomorphism $f_*: C_*(K) \to C_*(L)$ between the corresponding chain complexes
• The induced homomorphism $f_*$ is defined on the generators (oriented simplices) by: $f_*([v_0, \ldots, v_n]) = [f(v_0), \ldots, f(v_n)]$ and extended linearly to arbitrary chains
• The induced homomorphism $f_*$ commutes with the boundary operators, i.e., $f_* \circ \partial = \partial \circ f_*$

Induced homomorphisms on homology groups

• A simplicial map $f: K \to L$ also induces a homomorphism $f_*: H_*(K) \to H_*(L)$ between the corresponding homology groups
• The induced homomorphism on homology is defined by: $f_*([c]) = [f_*(c)]$ where $[c]$ denotes the homology class of a cycle $c$
• The induced homomorphism on homology is well-defined and captures the effect of the simplicial map on the "holes" of the simplicial complexes

Homotopy invariance

• Homotopy invariance is a fundamental property of homology that states that homology is preserved under continuous deformations (homotopies) of spaces
• In the simplicial setting, homotopy invariance is studied using simplicial maps and homotopy equivalence

Homotopic simplicial maps

• Two simplicial maps $f, g: K \to L$ are said to be homotopic if there exists a simplicial map $H: K \times I \to L$ (where $I$ is the simplicial complex representing the interval $[0, 1]$) such that:
  • $H(\cdot, 0) = f$
  • $H(\cdot, 1) = g$
• Intuitively, a homotopy between simplicial maps is a continuous deformation from one map to the other

Homotopy equivalence of simplicial complexes

• Two simplicial complexes $K$ and $L$ are said to be homotopy equivalent if there exist simplicial maps $f: K \to L$ and $g: L \to K$ such that:
  • $g \circ f$ is homotopic to the identity map on $K$
  • $f \circ g$ is homotopic to the identity map on $L$
• Homotopy equivalent simplicial complexes have the same topological properties, including homology groups

Invariance of homology under homotopy equivalence

• A key result in simplicial homology is that homotopy equivalent simplicial complexes have isomorphic homology groups
• If $K$ and $L$ are homotopy equivalent, then $H_n(K) \cong H_n(L)$ for all $n$
• This result allows us to study the homology of a space by considering any simplicial complex homotopy equivalent to it

Mayer-Vietoris sequence

• The Mayer-Vietoris sequence is a powerful tool in simplicial homology that relates the homology of a simplicial complex to the homology of its subspaces
• It provides a way to compute the homology of a space by decomposing it into simpler pieces

Decomposition of simplicial complexes

• Let $K$ be a simplicial complex, and let $L$ and $M$ be subcomplexes of $K$ such that $K = L \cup M$
• The Mayer-Vietoris sequence relates the homology of $K$ to the homology of $L$, $M$, and their intersection $L \cap M$

Mayer-Vietoris sequence

• The Mayer-Vietoris sequence is a long exact sequence of homology groups: $\cdots \to H_n(L \cap M) \xrightarrow{\alpha} H_n(L) \oplus H_n(M) \xrightarrow{\beta} H_n(K) \xrightarrow{\partial} H_{n-1}(L \cap M) \to \cdots$
• The maps $\alpha$, $\beta$, and $\partial$ are induced by the inclusions of the subcomplexes and the boundary operator
• The exactness of the sequence means that the kernel of each map is equal to the image of the previous map

Applications of Mayer-Vietoris sequence

• The Mayer-Vietoris sequence can be used to compute the homology of a simplicial complex by decomposing it into simpler subcomplexes
• For example, if $K$ is the union of two contractible subcomplexes $L$ and $M$, then the Mayer-Vietoris sequence implies that $H_n(K) \cong H_{n-1}(L \cap M)$ for all $n$
• The Mayer-Vietoris sequence is also useful for proving theoretical results, such as the homology of the sphere and the torus

Simplicial approximation theorem

• The simplicial approximation theorem is a fundamental result in algebraic topology that relates continuous maps between topological spaces to simplicial maps between their simplicial approximations
• It provides a way to study continuous maps using the combinatorial tools of simplicial complexes

Continuous maps vs simplicial maps

• A continuous map $f: X \to Y$ between topological spaces $X$ and $Y$ is a function that preserves the topological structure, i.e., the preimage of an open set is open
• A simplicial map $g: K \to L$ between simplicial complexes $K$ and $L$ is a function that preserves the combinatorial structure, i.e., it maps simplices to simplices
• The simplicial approximation theorem relates these two types of maps

Statement of simplicial approximation theorem

• Let $X$ and $Y$ be topological spaces, and let $K$ and $L$ be simplicial complexes such that $|K|$ is homeomorphic to $X$ and $|L|$ is homeomorphic to $Y$ (where $|K|$ and $|L|$ denote the geometric realizations of $K$ and $L$)
• For any continuous map $f: X \to Y$, there exists a simplicial map $g: K' \to L$, where $K'$ is a subdivision of $K$, such that $|g|$ is homotopic to $f$
• In other words, any continuous map can be approximated by a simplicial map up to homotopy, after sufficiently subdividing the domain simplicial complex

Applications of simplicial approximation theorem

• The simplicial approximation theorem allows us to study continuous maps between topological spaces using the combinatorial tools of simplicial complexes
• It is used to prove important results in algebraic topology, such as the Brouwer fixed point theorem and the Lefschetz fixed point theorem
• The theorem also provides a way to compute the induced homomorphisms on homology groups for continuous maps, by considering their simplicial approximations

Singular homology vs simplicial homology

• Singular homology and simplicial homology are two different approaches to defining homology for topological spaces
• While simplicial homology is defined using simplicial complexes, singular homology is defined using singular simplices, which are continuous maps from standard simplices to the space

Definition of singular homology

• Let $X$ be a topological space. A singular $n$-simplex in $X$ is a continuous map $\sigma: \Delta^n \to X$, where $\Delta^n$ is the standard $n$-simplex
• The singular chain complex $C_*(X)$ is defined as the free abelian group generated by singular simplices, with the boundary operator defined similarly to the simplicial case
• The singular homology groups $H_*(X)$ are defined as the homology groups of the singular chain complex $C_*(X)$

Comparison of singular and simplicial homology

• Singular homology is more general than simplicial homology, as it can be defined for any topological space, not just those that admit a simplicial structure
• For spaces that can be triangulated by a simplicial complex, the singular and simplicial homology groups are isomorphic
• The singular homology groups are homotopy invariant, meaning that homotopy equivalent spaces have isomorphic singular homology groups

Advantages and limitations of each approach

• Simplicial homology:
  • Advantages: Combinatorial nature, easier to compute in practice, provides a geometric