6.2 Singular homology and cohomology

Singular homology and cohomology are powerful tools for studying topological spaces. They use continuous maps from simplices to capture a space's structure, measuring "holes" and "obstructions" in different dimensions.

These theories satisfy key axioms like homotopy invariance and exactness. They're connected by the universal coefficient theorem, allowing us to compute one from the other in many cases. Understanding these concepts is crucial for analyzing complex topological spaces.

Singular Simplices and Chain Complexes

Definition and Notation

• A singular n-simplex in a topological space $X$ is a continuous map $\sigma: \Delta^n \to X$, where $\Delta^n$ is the standard n-simplex in $\mathbb{R}^{n+1}$
  • The standard n-simplex $\Delta^n$ is the convex hull of the standard basis vectors in $\mathbb{R}^{n+1}$ (e.g., the 1-simplex $\Delta^1$ is a line segment, the 2-simplex $\Delta^2$ is a triangle)
• The set of all singular n-simplices in $X$ is denoted by $S_n(X)$
• The singular chain complex of $X$ is the sequence of free abelian groups $C_n(X)$ generated by the singular n-simplices, with boundary maps $\partial_n: C_n(X) \to C_{n-1}(X)$
  • The boundary maps are defined by the alternating sum of the faces of each simplex, where the i-th face of a simplex $\sigma$ is obtained by omitting the i-th vertex

Properties and Interpretation

• The boundary maps satisfy the property $\partial_{n-1} \circ \partial_n = 0$, making the singular chain complex a chain complex
  • This property ensures that the composition of two consecutive boundary maps is always zero, which is crucial for defining homology groups
• The singular chain complex encodes the topological information of the space $X$ through the continuous maps from simplices to $X$
  • Intuitively, the singular chain complex captures how the space $X$ can be "triangulated" or approximated by simplices of various dimensions
  • The boundary maps describe how these simplices are connected to each other, with the boundary of an n-simplex being a combination of (n-1)-simplices

Singular Homology and Cohomology Groups

Definition and Interpretation

• The singular homology groups $H_n(X)$ are defined as the homology groups of the singular chain complex, i.e., $H_n(X) = \ker(\partial_n) / \operatorname{im}(\partial_{n+1})$
  • Elements of $H_n(X)$ are equivalence classes of n-cycles (singular n-chains with zero boundary) modulo n-boundaries (singular n-chains that are the boundary of an (n+1)-chain)
  • Intuitively, homology groups measure the "holes" or "voids" in a space $X$ that cannot be filled by simplices of the corresponding dimension (e.g., $H_0(X)$ measures the connected components of $X$, $H_1(X)$ measures the 1-dimensional holes or loops)
• The singular cohomology groups $H^n(X; G)$ are defined as the cohomology groups of the singular cochain complex, which is the dual of the singular chain complex
  • The singular cochain complex is obtained by applying the $\operatorname{Hom}$ functor $\operatorname{Hom}(-, G)$ to the singular chain complex, where $G$ is an abelian group (usually $\mathbb{Z}$ or a field)
  • The coboundary maps $\delta^n: C^n(X; G) \to C^{n+1}(X; G)$ are the dual maps of the boundary maps, satisfying $\delta^{n+1} \circ \delta^n = 0$
  • Elements of $H^n(X; G)$ are equivalence classes of n-cocycles (singular n-cochains with zero coboundary) modulo n-coboundaries (singular n-cochains that are the coboundary of an (n-1)-cochain)

Relationship between Homology and Cohomology

• Singular homology and cohomology are related by the universal coefficient theorem, which states that for a topological space $X$ and an abelian group $G$, there is a short exact sequence: $0 \to \operatorname{Ext}(H_{n-1}(X), G) \to H^n(X; G) \to \operatorname{Hom}(H_n(X), G) \to 0$
  • This sequence splits (non-naturally), so $H^n(X; G) \cong \operatorname{Ext}(H_{n-1}(X), G) \oplus \operatorname{Hom}(H_n(X), G)$
  • When $G$ is a field or $H_{n-1}(X)$ is free, $\operatorname{Ext}(H_{n-1}(X), G) = 0$, and $H^n(X; G) \cong \operatorname{Hom}(H_n(X), G)$
• Intuitively, cohomology groups can be thought of as dual to homology groups, measuring the "co-holes" or "obstructions" in a space $X$
  • While homology groups are computed using cycles and boundaries, cohomology groups are computed using cocycles and coboundaries, which are dual notions

Axioms for Homology and Cohomology Theories

Homotopy Invariance and Exactness

• Homotopy invariance: If $f, g: X \to Y$ are homotopic continuous maps, then the induced homomorphisms $f_*, g_*: H_n(X) \to H_n(Y)$ and $f^*, g^*: H^n(Y; G) \to H^n(X; G)$ are equal for all $n$
  • This axiom states that homology and cohomology groups are invariant under homotopy equivalence, capturing the idea that they depend only on the "shape" of a space, not on its specific embedding or parametrization
• Exactness: For any pair of topological spaces $A \subset X$, there are long exact sequences in both homology and cohomology connecting the groups of $A$, $X$, and $X/A$:
  • $\cdots \to H_n(A) \to H_n(X) \to H_n(X, A) \to H_{n-1}(A) \to \cdots$
  • $\cdots \to H^n(X, A; G) \to H^n(X; G) \to H^n(A; G) \to H^{n+1}(X, A; G) \to \cdots$
  • These sequences relate the homology and cohomology groups of a space $X$, a subspace $A$, and the quotient space $X/A$, providing a powerful tool for computing these groups in terms of simpler spaces

Excision and Additivity

• Excision: If $U \subset A \subset X$ and the closure of $U$ is contained in the interior of $A$, then the inclusion $(X - U, A - U) \hookrightarrow (X, A)$ induces isomorphisms $H_n(X - U, A - U) \cong H_n(X, A)$ and $H^n(X, A; G) \cong H^n(X - U, A - U; G)$ for all $n$
  • This axiom allows the computation of relative homology and cohomology groups by "excising" a subset $U$ that is contained in the interior of $A$, without changing the groups
  • It is a crucial tool for proving the Mayer-Vietoris sequence, which relates the homology and cohomology groups of a space $X$ to those of two subspaces $A$ and $B$ whose union is $X$
• Additivity: For any disjoint union of topological spaces $X = \bigsqcup_\alpha X_\alpha$, there are natural isomorphisms $H_n(X) \cong \bigoplus_\alpha H_n(X_\alpha)$ and $H^n(X; G) \cong \prod_\alpha H^n(X_\alpha; G)$ for all $n$
  • This axiom states that homology and cohomology groups of a disjoint union of spaces are the direct sum and direct product, respectively, of the groups of the individual spaces
  • It allows the computation of homology and cohomology groups of a space by decomposing it into simpler pieces

Dimension Axiom

• Dimension axiom: For a one-point space $\{*\}$, the homology and cohomology groups are given by:
  • $H_n(\{*\}) \cong \begin{cases} \mathbb{Z}, & n = 0 \\ 0, & n \neq 0 \end{cases}$
  • $H^n(\{*\}; G) \cong \begin{cases} G, & n = 0 \\ 0, & n \neq 0 \end{cases}$
  • This axiom specifies the homology and cohomology groups of a single point, which serve as the building blocks for the groups of more complex spaces via the other axioms
  • It also ensures that the homology and cohomology theories are non-trivial, as they assign non-zero groups to the simplest possible space

Singular Homology and Cohomology Computations

Basic Topological Spaces

• For a contractible space $X$ (e.g., a point, a ball, or a convex set in $\mathbb{R}^n$), $H_n(X) \cong H^n(X; G) \cong G$ for $n = 0$ and $0$ for $n \neq 0$
  • Contractible spaces have the same homology and cohomology groups as a single point, as they can be continuously deformed to a point without changing their topology
• For the n-sphere $S^n$, the homology and cohomology groups are given by:
  • $H_0(S^n) \cong H_n(S^n) \cong \mathbb{Z}$, $H_0(S^n; G) \cong H_n(S^n; G) \cong G$, and $H_k(S^n) \cong H_k(S^n; G) \cong 0$ for $0 < k < n$
  • The n-sphere has non-trivial homology and cohomology groups only in dimensions $0$ and $n$, reflecting the fact that it is a connected space with a single "hole" or "void" in dimension $n$

Operations on Topological Spaces

• For a wedge sum of spaces $X \vee Y$, there are natural isomorphisms $H_n(X \vee Y) \cong H_n(X) \oplus H_n(Y)$ and $H^n(X \vee Y; G) \cong H^n(X; G) \times H^n(Y; G)$ for all $n$, provided that the wedge point is the only common point of $X$ and $Y$
  • The wedge sum of two spaces is obtained by gluing them together at a single point, and its homology and cohomology groups are the direct sum and direct product, respectively, of the groups of the individual spaces
• For a product of spaces $X \times Y$, there are natural isomorphisms:
  • $H_n(X \times Y) \cong \bigoplus_{i+j=n} H_i(X) \otimes H_j(Y)$ (Künneth formula for homology)
  • $0 \to \bigoplus_{i+j=n} H^i(X; G) \otimes H^j(Y; G) \to H^n(X \times Y; G) \to \bigoplus_{i+j=n-1} \operatorname{Tor}(H^i(X; G), H^j(Y; G)) \to 0$ (Künneth formula for cohomology)
  • These formulas relate the homology and cohomology groups of a product space to the tensor products and torsion products of the groups of the individual spaces
• For a quotient space $X/A$, where $A$ is a contractible subspace of $X$, there are natural isomorphisms $H_n(X/A) \cong H_n(X, A)$ and $H^n(X/A; G) \cong H^n(X, A; G)$ for all $n$ (relative homology and cohomology)
  • These isomorphisms allow the computation of the homology and cohomology groups of a quotient space by considering the relative groups of the original space and the subspace being collapsed
  • They are particularly useful when the subspace $A$ is contractible, as the relative groups $(X, A)$ are easier to compute than the groups of the quotient space $X/A$ directly