Introduction to Algebraic Topology

9.1 Homotopy

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).

9.2 The Fundamental Group

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.

9.3 Directly Connected Spaces

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.

9.4 Euler Characteristic

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 $g$: $\chi = 2 - 2g$.

9.5 Classification of Surfaces

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 $\mathbb{R}^3$.