Introduction to Algebraic Topology

Definition. Two continuous functions $f, g : X \to Y$ are homotopic (written $f \simeq g$) if there exists a continuous map $H : X \times [0, 1] \to Y$ such that $H(x, 0) = f(x)$ and $H(x, 1) = g(x)$ for all $x \in X$.

The map $H$ is called a homotopy from $f$ to $g$.

Definition. Two spaces $X$ and $Y$ are homotopy equivalent (written $X \simeq Y$) if there exist continuous maps $f : X \to Y$ and $g : Y \to X$ such that $g \circ f \simeq \operatorname{id}_X$ and $f \circ g \simeq \operatorname{id}_Y$.

Example 9.1. The solid disc $D^2 = \{(x, y) : x^2 + y^2 \leq 1\}$ is homotopy equivalent to the single point $\{0\}$ (it is contractible).

Definition. A loop in $X$ based at $x_0 \in X$ is a continuous map $\gamma : [0, 1] \to X$ with $\gamma(0) = \gamma(1) = x_0$.

Definition. The fundamental group $\pi_1(X, x_0)$ is the set of homotopy classes of loops based at $x_0$, with the group operation given by concatenation of loops.

For path-connected spaces, $\pi_1(X, x_0)$ is independent of the choice of basepoint $x_0$ (up to isomorphism).

Proposition 9.1. The fundamental group is a topological invariant: if $X \cong Y$, then $\pi_1(X) \cong \pi_1(Y)$.

Example 9.2. $\pi_1(S^1) \cong \mathbb{Z}$. From first principles, loops in $S^1$ are classified by their winding number.

Example 9.3. $\pi_1(\mathbb{R}^n) = \{e\}$ (the trivial group) for all $n \geq 1$. More generally, the fundamental group of any directly connected space is trivial.

Example 9.4. $\pi_1(T^2) \cong \mathbb{Z} \times \mathbb{Z}$, where $T^2$ is the torus.

Definition. A path-connected space $X$ is directly connected if $\pi_1(X) \cong \{e\}$.

Equivalently, every loop in $X$ can be continuously contracted to a point.

Proposition 9.2. $\mathbb{R}^n$, $S^n$ (for $n \geq 2$), and any convex subset of $\mathbb{R}^n$ are directly connected.

Definition. For a finite CW-complex (e.g., a polyhedron), the Euler characteristic is:

$\chi = V - E + F$

where $V$ = number of vertices, $E$ = number of edges, $F$ = number of faces (or higher-dimensional cells more generally).

Example 9.5.

For a closed orientable surface of genus $g$: $\chi = 2 - 2g$.

Theorem 9.1 (Classification of Compact Surfaces). Every compact connected surface is homeomorphic to exactly one of:

1. A sphere with $g$ handles (orientable, genus $g$), or
2. A sphere with $g$ cross-caps / Möbius bands (non-orientable, genus $g$).

Key surfaces:

• Torus $T^2$: A coffee mug / donut. Constructed by identifying opposite edges of a square.
• Projective plane $\mathbb{R}P^2$: Obtained by identifying antipodal points of $S^2$. Non-orientable.
• Klein bottle $K$: Obtained by identifying opposite edges of a square with one pair reversed. Non-orientable, cannot be embedded in $\mathbb{R}^3$.

Definition. A continuous surjection $p : \tilde{X} \to X$ is a covering map if every point $x \in X$ has an open neighbourhood $U$ such that $p^{-1}(U)$ is a disjoint union of open sets in $\tilde{X}$, each mapped homeomorphically onto $U$ by $p$.

Example 9.6. The map $p : \mathbb{R} \to S^1$ defined by $p(t) = e^{2\pi i t}$ is a covering map. The preimage of any small arc on $S^1$ consists of infinitely many disjoint intervals in $\mathbb{R}$.

Theorem 9.2 (Lifting Property). If $p : \tilde{X} \to X$ is a covering map, $\tilde{X}$ is path-connected, and $f : [0, 1] \to X$ is a loop based at $x_0$, then $f$ lifts to a unique path $\tilde{f} : [0, 1] \to \tilde{X}$ starting at any chosen preimage $\tilde{x}_0 \in p^{-1}(x_0)$.

The fundamental group $\pi_1(S^1) \cong \mathbb{Z}$ can be proved using this lifting property: a loop in $S^1$ lifts to a path in $\mathbb{R}$ whose endpoints differ by an integer, the winding number.

• Homotopy equivalence is weaker than homeomorphism: Two spaces can be homotopy equivalent without being homeomorphic (e.g., $\mathbb{R}^n$ and a point are homotopy equivalent but not homeomorphic).
• The fundamental group detects “holes”: A space with trivial $\pi_1$ has no one-dimensional holes; non-trivial $\pi_1$ indicates loops that cannot be contracted.
• Euler characteristic is a homotopy invariant: Any two homotopy equivalent spaces have the same Euler characteristic, even though it can be computed from any triangulation.
• Covering spaces relate local and global topology: The universal cover of a space $X$ is simply connected, and $\pi_1(X)$ acts on it as deck transformations.
• The classification of surfaces reduces topology to algebra: Every compact surface is determined by its orientability and Euler characteristic.

• Confusing homotopy with homeomorphism: Two spaces can be homotopy equivalent without being homeomorphic (e.g., a solid disc and a point). Homotopy equivalence is a much coarser relation than homeomorphism.
• Assuming $\pi_1(X)$ is always abelian: The fundamental group of a general space need not be abelian. For example, $\pi_1(S^1 \vee S^1)$ is the free group on two generators, which is non-abelian.
• Forgetting basepoint dependence: For non-path-connected spaces, the fundamental group depends on the choice of basepoint, and different components may have different fundamental groups.

• Robotics and motion planning: The fundamental group of a configuration space determines whether paths between configurations can be continuously deformed into each other.
• Data analysis (Topological Data Analysis): Persistent homology detects topological features (connected components, loops, voids) in high-dimensional data sets.
• Physics: Homotopy groups classify topological defects in condensed matter (e.g., vortices in superfluids, dislocations in crystals).
• Knot theory: The fundamental group of the knot complement is a powerful knot invariant used to distinguish different knots.