Reference

The Whole Story on One Page

The key definitions, theorems and computations of Parts III to V, in the order they appear — each with a link back to its chapter.

Chains and boundaries

An oriented $k$ -simplex $[v_{0}, \dots, v_{k}] .$ A $k$ -chain is a formal sum of $k$ -simplices with coefficients in $G :$

C_{k} (K; G) = {\sum_{i} g_{i} σ_{i}} .

The boundary operator:

\partial [v_{0}, \dots, v_{k}] = i = 0 \sum k (- 1)^{i} [v_{0}, \dots, v^_{i}, \dots, v_{k}], \partial \circ \partial = 0.

§3.2 Chains and the Boundary Operator

Homology

Cycles are chains with no boundary; boundaries are boundaries of something one dimension up:

Z_{k} B_{k} H_{k} = ker \partial_{k}, = im \partial_{k + 1} \subseteq Z_{k}, = Z_{k} / B_{k} .

Betti numbers (over a field $F),$ with $n_{k}$ the number of $k$ -simplices:

b_{k} = dim H_{k} = n_{k} - rank \partial_{k} - rank \partial_{k + 1} .

§3.3 Homology Groups

Euler–Poincaré

The alternating count of cells equals the alternating sum of Betti numbers:

χ (X) = k \sum (- 1)^{k} n_{k} = k \sum (- 1)^{k} b_{k} .

For closed surfaces: $χ (Σ_{g}) = 2 - 2 g$ (orientable, genus $g);$ $χ = 2 - k$ for a sum of $k$ projective planes.

§3.3 Homology Groups

Torsion

Every finitely generated abelian group is

Z^{r} \oplus Z / d_{1} \oplus \dots \oplus Z / d_{m}, with d_{1} ∣ d_{2} ∣ \dots ∣ d_{m} .

The free rank is read from ranks of boundary matrices; the torsion $Z / d_{i}$ of $H_{k}$ from the Smith normal form of $\partial_{k + 1}$ (diagonal entries $d_{i} > 1).$

§3.4 Computing Homology

Invariance and functoriality

A continuous map $f : X \to Y$ induces $f_{*} : H_{k} (X) \to H_{k} (Y),$ with

(g \circ f)_{*} = g_{*} \circ f_{*}, (id_{X})_{*} = id, f ≃ g \Rightarrow f_{*} = g_{*} .

So homotopy-equivalent spaces have isomorphic homology. Payoffs: Brouwer’s fixed point theorem, the hairy ball theorem, invariance of dimension.

§3.5 Maps, Invariance, and First Triumphs

Exact sequences

Mayer–Vietoris, for $X = U \cup V$ with $U, V$ open:

\dots \to H_{k} (U \cap V) \to H_{k} (U) \oplus H_{k} (V) \to H_{k} (X) \partial_{*} H_{k - 1} (U \cap V) \to \dots

Every map’s image is exactly the next map’s kernel.

§3.6 Exact Sequences and Mayer–Vietoris

Knots

For every knot $K,$ Mayer–Vietoris gives $H_{1} (S^{3} ∖ K) ≅ Z,$ generated by the meridian $μ;$ the class of a loop is its linking number with $K :$

[γ] = lk (γ, K) [μ] .

Finer invariants: the Alexander polynomial from a Seifert matrix $V,$ the Jones polynomial from the Kauffman bracket of a diagram $D,$ and Khovanov homology, whose graded Euler characteristic is the Jones polynomial:

Δ_{K} (t) ≐ det (V - t V^{T}), V_{K} (t) = (- A^{3})^{- w (D)} ⟨ D ⟩, A = t^{- 1/4}, \sum_{i, j} (- 1)^{i} q^{j} dim Kh^{i, j} (K) = (q + q^{- 1}) V_{K} (q^{2}) .

§3.7 Knots and Links

Persistence

Grow balls of radius $r$ around the points of a cloud $P .$ The Čech complex is their nerve and has the homology of their union $U_{r};$ the Vietoris–Rips complex only asks the balls to meet in pairs:

C ˇ_{r} (P) \subseteq VR_{r} (P) \subseteq C ˇ_{2 r} (P) .

As $r$ grows, each class is born and dies; the intervals $[b, d)$ form the barcode. Stability: moving every point by at most $δ$ moves the diagrams by at most $δ,$

d_{B} (D_{k} (P), D_{k} (P^{'})) \leq δ .

§3.8 Persistent Homology

Cohomology

Cochains measure chains; the coboundary is the transpose of the boundary:

C^{k} (K; G) = Hom (C_{k} (K), G), (δ φ) (σ) = φ (\partial σ), δ \circ δ = 0, H^{k} = ker δ^{k} / im δ^{k - 1} .

Maps pull back: $f : X \to Y$ gives $f^{*} : H^{k} (Y) \to H^{k} (X) .$

§4.2 Cohomology Groups

Universal coefficients

Cohomology is determined by homology, but torsion moves up one degree:

H^{k} (X; Z) ≅ Hom (H_{k} (X), Z) \oplus Ext (H_{k - 1} (X), Z) .

Example: $H_{1} (RP^{2}) = Z /2$ and $H_{2} (RP^{2}) = 0,$ yet $H^{2} (RP^{2}; Z) = Z /2 .$

§4.2 Cohomology Groups

de Rham and Stokes

\int_{\partial M} ω = \int_{M} d ω, d \circ d = 0, H_{dR}^{k} (M) = \frac{{ closed k -forms }}{{ exact k -forms }} ≅ H^{k} (M; R) .

On the punctured plane, $d θ = (- y d x + x d y) / (x^{2} + y^{2})$ is closed but not exact: $\oint_{γ} d θ = 2 π \cdot (winding number of γ) .$

§4.4 de Rham Cohomology

Cup product

(α ⌣ β) (σ) α ⌣ β = α (σ ∣_{[v_{0}, \dots, v_{p}]}) \cdot β (σ ∣_{[v_{p}, \dots, v_{p + q}]}), = (- 1)^{pq} β ⌣ α .

$T^{2}$ and $S^{1} \lor S^{1} \lor S^{2}$ have the same groups; on $T^{2},$ $a ⌣ b$ generates $H^{2},$ on the wedge every such product is $0 .$

§4.5 The Cup Product

Poincaré duality

For a closed, oriented $n$ -manifold $M :$

H^{k} (M; Z) ≅ H_{n - k} (M; Z), b_{k} = b_{n - k} .

With $Z /2$ coefficients it holds for every closed manifold, orientable or not.

§4.6 Poincaré Duality

Sheaves and Čech cohomology

Cover $X$ by open sets $U$ and do cohomology on the overlaps. For a good cover the nerve has the homotopy type of $X,$ so

H ˇ^{p} (U; G) ≅ H^{p} (X; G) .

For a sheaf $F$ of local data, $H ˇ^{0} = F (X)$ is the global sections: local solutions that agree on overlaps. $H ˇ^{1}$ measures how local solutions fail to glue.

§4.7 Sheaves and Čech Cohomology

Gauss–Bonnet

Total curvature of a closed surface is topology:

\int_{M} K d A = 2 π χ (M) .

Sphere: $4 π .$ Torus: $0 .$ The hairy ball theorem: a sphere has no nowhere-zero tangent vector field, because $χ (S^{2}) = 2 \neq = 0 .$

§4.8 Curvature and Characteristic Classes

Categories and functors

A functor $F : C \to D$ sends objects to objects and arrows to arrows, keeping identities and composition; homology $H_{n} : Top \to Ab$ is one. A natural transformation $η : F \Rightarrow G$ has components $η_{X} : F (X) \to G (X)$ with

G (f) \circ η_{X} = η_{Y} \circ F (f)

for every arrow $f : X \to Y :$ the naturality square commutes.

§5.1 Categories and Functors

The axioms

Eilenberg and Steenrod: a homology theory is a functor on pairs with connecting maps $\partial,$ satisfying homotopy, exactness, excision, dimension and additivity. Given the group of a point, these five axioms determine it on finite complexes; with $H_{0} (pt) = Z,$

H_{n} (S^{k}) ≅ {Z 0 n = 0 or n = k, otherwise (k \geq 1) .

Drop the dimension axiom and you get the generalized theories, such as bordism and K-theory.

§5.2 Homological Algebra

Hodge decomposition

Every flow $f$ on the edges of a complex splits, perpendicularly, into a gradient, a curl and a harmonic part:

f = δ_{0} s + δ_{1}^{T} φ + h, ∥ f ∥^{2} = ∥ δ_{0} s ∥^{2} + ∥ δ_{1}^{T} φ ∥^{2} + ∥ h ∥^{2} .

Each class of $H^{1} (K; R)$ contains exactly one harmonic flow, so they form a space of dimension $b_{1} .$ For forms on a closed manifold, $Ω^{k} = d Ω^{k - 1} \oplus d^{*} Ω^{k + 1} \oplus H^{k}$ with $dim H^{k} = b_{k} .$

§5.3 Horizons

Homology of famous spaces (integer coefficients)

Space	$H_{0}$	$H_{1}$	$H_{2}$	higher	$χ$
point	$Z$	$0$	$0$	$0$	$1$
circle $S^{1}$	$Z$	$Z$	$0$	$0$	$0$
wedge of $n$ circles	$Z$	$Z^{n}$	$0$	$0$	$1 - n$
sphere $S^{2}$	$Z$	$0$	$Z$	$0$	$2$
sphere $S^{n},$ $n \geq 1$	$Z$	$Z$ in degree $n,$ $0$ otherwise			$1 + (- 1)^{n}$
torus $T^{2}$	$Z$	$Z^{2}$	$Z$	$0$	$0$
genus- $g$ surface $Σ_{g}$	$Z$	$Z^{2 g}$	$Z$	$0$	$2 - 2 g$
Klein bottle $K$	$Z$	$Z \oplus Z /2$	$0$	$0$	$0$
projective plane $RP^{2}$	$Z$	$Z /2$	$0$	$0$	$1$
Möbius band	$Z$	$Z$	$0$	$0$	$0$
$n$ -torus $T^{n}$	$Z$	$Z^{n}$	$Z^{(2 n)}$	$H_{k} = Z^{(k n)}$	$0$
$CP^{n}$	$Z$	$0$	$Z$	$Z$ in even degrees $\leq 2 n$	$n + 1$

With $Z /2$ coefficients the Klein bottle has Betti numbers $1, 2, 1$ — the same as the torus — and $RP^{2}$ has $1, 1, 1 .$ Torsion is exactly what such coefficients cannot see.