Algebraic Topology

Fundamental Group π₁(X, x₀)

Definition

Group of loops at $x_0$ under homotopy.

$$\pi_1(X, x_0) = \{[\gamma] : \gamma \text{ loop at } x_0\}$$

Path Homotopy

$f \simeq g$ rel $\{0,1\}$ if $H: I \times I \to X$ with $H(t,0)=f(t)$, $H(t,1)=g(t)$

Operation: $[\gamma_1] \cdot [\gamma_2] = [\gamma_1 * \gamma_2]$ (concatenation)

Key Computations

$\pi_1(\text{Disk}) = \{e\}$ (contractible)

$\pi_1(S^1) \cong \mathbb{Z}$ (winding number)

$\pi_1(S^n) = \{e\}$ for $n \geq 2$

$\pi_1(T^n) \cong \mathbb{Z}^n$ (product)

The fundamental group measures "how many holes" a space has in 1D.

Covering Spaces

Definition

$p: \tilde{X} \to X$ is covering if every $x \in X$ has nbhd $U$ with $p^{-1}(U)$ disjoint union of open sets homeomorphic to $U$.

Lifting Property

If $p: \tilde{X} \to X$ covering, $\gamma: I \to X$ path, $\tilde{x}_0 \in p^{-1}(\gamma(0))$, then $\exists! \tilde{\gamma}$ with $p \circ \tilde{\gamma} = \gamma$ and $\tilde{\gamma}(0) = \tilde{x}_0$

Deck Transformations

$f: \tilde{X} \to \tilde{X}$ homeomorphism with $p \circ f = p$

$\text{Deck}(\tilde{X}/X) \cong \pi_1(X)/p_*(\pi_1(\tilde{X}))$

Universal Cover

Simply connected covering space $\tilde{X}$ of $X$

Unique up to homeomorphism over $X$

Not all spaces have universal covers (must be path-connected, locally path-connected)

Seifert-Van Kampen Theorem

Statement

If $X = U \cup V$ with $U, V$ open, path-connected, $U \cap V$ path-connected:

$$\pi_1(X) \cong \pi_1(U) *_{\pi_1(U \cap V)} \pi_1(V)$$

Amalgamated Product

Generators: all generators from $\pi_1(U)$ and $\pi_1(V)$

Relations: all relations from both + identification from $\pi_1(U \cap V)$

Application

Compute $\pi_1(X)$ by breaking into simpler pieces

Example: $\pi_1(S^1 \vee S^1) \cong \mathbb{Z} * \mathbb{Z}$ (free group on 2 generators)

Powerful for computing fundamental groups of spaces that are unions of simpler spaces.

Simplicial Complexes

Definition

Collection $K$ of simplices where: (1) each face in $K$, (2) intersection of two is face of both.

Simplices

$n$-simplex: convex hull of $n+1$ affinely indep. points

0-simplex: point; 1-simplex: edge; 2-simplex: triangle

Face: subset of vertices (all faces included)

Triangulation

Decompose space $X$ into simplices: $|K| = X$

Geometric realization: $|K|$ is union of simplices with prescribed topology

$\chi(K) = \sum (-1)^i n_i$ (Euler characteristic)

Simplicial complexes provide combinatorial models for topological spaces.

Singular Homology

Definition

$\sigma: \Delta^n \to X$ continuous (singular $n$-simplex)

$C_n(X) = $ free abelian group generated by singular $n$-simplices

Boundary Map

$$\partial_n(\sigma) = \sum_{i=0}^n (-1)^i \sigma|_{[v_0,...,\hat{v_i},...,v_n]}$$

$\partial_n \circ \partial_{n+1} = 0$ (chain complex)

Chain Complex

$$... \xrightarrow{\partial_{n+1}} C_n(X) \xrightarrow{\partial_n} C_{n-1}(X) \xrightarrow{\partial_{n-1}} ...$$

Cycles & Boundaries

$Z_n = \ker(\partial_n)$ (cycles)

$B_n = \text{im}(\partial_{n+1})$ (boundaries)

$H_n(X) = Z_n / B_n$ (homology)

Boundary of boundary is zero: $\partial^2 = 0$

Homology Groups H_n(X)

Interpretation

$H_0$: path components

$H_1$: 1D holes (loops mod boundaries)

$H_n$: $n$-dimensional holes

Betti Numbers

$b_n = \text{rank}(H_n(X; \mathbb{Q}))$

$$\chi(X) = \sum (-1)^n b_n$$

Key Examples

$H_n(S^k) = \mathbb{Z}$ for $n=0,k$; $0$ else

$H_n(T^k) = \binom{k}{n}\mathbb{Z}$

$H_1(X) \cong \pi_1(X)^{ab}$ (abelianization)

Homology detects topological features algebraically.

Exact Sequences

Short Exact Sequence

$$0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$$

$f$ injective, $g$ surjective, $\text{im}(f) = \ker(g)$

Long Exact Sequence

Chains of homomorphisms where $\text{im}(f_n) = \ker(f_{n-1})$ for all $n$

Snake Lemma

Gives connecting homomorphism $\partial: \ker(g) \to \text{cok}(f)$

$$... \to H_n(A) \xrightarrow{f_*} H_n(B) \xrightarrow{g_*} H_n(C) \xrightarrow{\partial} H_{n-1}(A) \to ...$$

Exact sequences are powerful computational tools—5-lemma, diagram chasing.

Mayer-Vietoris Sequence

Setup

$X = U \cup V$, $U, V$ open, with $U \cap V \neq \emptyset$:

Long Exact Sequence

$$... \to H_n(U \cap V) \to H_n(U) \oplus H_n(V) \to H_n(X) \xrightarrow{\partial} H_{n-1}(U \cap V) \to ...$$

Application

Compute $H_n(X)$ from pieces $U$, $V$, $U \cap V$

Example: $H_*(S^n)$ via upper/lower hemispheres

Combine with induction on CW complex structure.

Relative Homology H_n(X, A)

Definition

Quotient complex: $C_n(X, A) = C_n(X) / C_n(A)$

$$H_n(X, A) = \ker(\partial_n) / \text{im}(\partial_{n+1})$$

Long Exact Sequence

$$... \to H_n(A) \to H_n(X) \to H_n(X, A) \xrightarrow{\partial} H_{n-1}(A) \to ...$$

Excision Axiom

If $Z \subset A \subset X$ and $\bar{Z} \subset \text{int}(A)$:

$$H_n(X, A) \cong H_n(X \setminus Z, A \setminus Z)$$

Interpretation

$H_n(X, A)$ measures "holes in $X$ relative to $A$"

Remove null-homotopic parts without changing homology.

Cohomology H^n(X)

Definition

Dual of chain complex: $C^n(X) = \text{Hom}(C_n(X), R)$

Coboundary: $\delta^n = (\partial_{n+1})^*$

Cochain Complex

$$... \xrightarrow{\delta^{n-1}} C^n(X) \xrightarrow{\delta^n} C^{n+1}(X) \xrightarrow{\delta^{n+1}} ...$$

$$H^n(X) = \ker(\delta^n) / \text{im}(\delta^{n-1})$$

Pontryagin Duality

$H^n(X; R) \cong \text{Hom}(H_n(X), R) \oplus \text{Ext}(H_{n-1}(X), R)$ (for $R = \mathbb{Z}$)

$H^n(X; \mathbb{Q}) \cong \text{Hom}(H_n(X; \mathbb{Q}), \mathbb{Q})$ (for fields)

Cohomology is contravariant; naturality with maps $f: X \to Y$.

Cup Product

Definition

$\cup: H^p(X) \times H^q(X) \to H^{p+q}(X)$

Cohomology becomes graded ring with $\cup$ product

Properties

Associative: $(a \cup b) \cup c = a \cup (b \cup c)$

Commutative (up to sign): $a \cup b = (-1)^{pq} b \cup a$

Identity element in $H^0$

Ring Structure

$H^*(X) = \bigoplus_n H^n(X)$ forms graded ring

Functor: $f^*: H^*(Y) \to H^*(X)$ is ring homomorphism

$$f^*(a \cup b) = f^*(a) \cup f^*(b)$$

Cup product detects additional structure beyond homology.

Poincaré Duality

Theorem

For closed, orientable $n$-manifold $M$:

$$H_k(M) \cong H^{n-k}(M)$$

With Coefficients

$H_k(M; R) \cong H^{n-k}(M; R)$

Via cap product: $[M] \cap (-): H^{n-k}(M) \to H_k(M)$

Cap Product

$$\cap: H_n(X) \times H^k(X) \to H_{n-k}(X)$$

Non-orientable & Relative

Use $\mathbb{Z}/2$ coefficients for non-orientable

Relative version: $H_k(M, \partial M) \cong H^{n-k}(M)$

Requires manifold structure and orientability (or $\mathbb{Z}/2$ coefficients).

Euler Characteristic χ(X)

Definition

$$\chi(X) = \sum_{n=0}^\infty (-1)^n b_n$$

where $b_n = \text{rank}(H_n(X; \mathbb{Q}))$

Combinatorial Formula

For cell complex: $\chi(X) = \sum (-1)^n c_n$ (number of $n$-cells)

For simplicial complex: $\chi(K) = V - E + F - ...$

Key Examples

$\chi(S^n) = 1 + (-1)^n$ (even $n$: 2; odd $n$: 0)

$\chi(T^2) = 0$; $\chi(\mathbb{RP}^2) = 1$

Additivity

$\chi(X \cup Y) = \chi(X) + \chi(Y) - \chi(X \cap Y)$

$\chi(X \times Y) = \chi(X) \cdot \chi(Y)$

Invariant under homotopy equivalence.

CW Complexes

Definition

Built by attaching $n$-cells via maps $S^{n-1} \to X^{(n-1)}$

$X^{(n)}$ = $n$-skeleton (union of cells of dim $\leq n$)

Weak topology: $U$ open iff $U \cap X^{(n)}$ open for all $n$

Cellular Chain Complex

$$C_n^{CW}(X) = \mathbb{Z}[\text{$n$-cells}]$$

$H_n^{CW}(X) \cong H_n(X)$ (cellular homology)

Examples

$S^n$: one 0-cell, one $n$-cell

$\mathbb{RP}^n$: one cell per dimension

$T^2$: one 0-cell, two 1-cells, one 2-cell

CW structures simplify homology computation dramatically.

Cellular Homology

Setup

For CW complex $X$, relative homology chain complex:

$$C_n^{CW}(X) = H_n(X^{(n)}, X^{(n-1)})$$

Boundary Map

Computed from attaching maps of $(n+1)$-cells

Degree of attaching map determines boundary coefficient

Theorem

$H_n^{CW}(X) = H_n(X)$ (cellular homology equals singular homology)

Computation

Finite dimensional for finite CW complexes

Matrix of boundary map is $|n\text{-cells}| \times |(n-1)\text{-cells}|$

Practical algorithm: compute boundaries of cells, find rank & nullity.

Higher Homotopy Groups π_n(X)

Definition

Maps $S^n \to X$ up to homotopy (rel basepoint)

$$\pi_n(X, x_0) = [S^n, X]_{x_0}$$

Abelian for n ≥ 2

$\pi_n(X)$ is abelian group (under concatenation)

$\pi_1(X)$ acts on $\pi_n(X)$ for $n \geq 2$

Key Facts

$\pi_n(S^k) = 0$ if $n < k$ (contractible)

$\pi_n(S^n) \cong \mathbb{Z}$ (degree)

$\pi_n(S^k)$ finite for $n > k$

Hurewicz Theorem

If $\pi_i(X) = 0$ for $i < n$, then $\pi_n(X) \cong H_n(X)$

Higher homotopy groups are harder to compute than homology.