4.1 Homotopy of maps and homotopy equivalence

Homotopy of maps and homotopy equivalence are key concepts in algebraic topology. They provide a way to compare continuous functions and topological spaces, allowing us to group similar objects and study their shared properties.

These ideas form the foundation for understanding the fundamental group and other homotopy groups. By focusing on continuous deformations, we can explore the essential structure of spaces and maps, leading to powerful classification tools in topology.

Homotopy of Maps

Definition and Notation

• A homotopy between two continuous maps $f, g: X \to Y$ is 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 interval $[0, 1]$ represents the parameter space for the homotopy
  • The homotopy continuously deforms the map $f$ into the map $g$ as the parameter varies from $0$ to $1$
• If a homotopy exists between $f$ and $g$, then $f$ and $g$ are said to be homotopic, denoted by $f \simeq g$
• The set of homotopy classes of maps from $X$ to $Y$ is denoted by $[X, Y]$

Properties of Homotopy

• The homotopy relation is an equivalence relation on the set of continuous maps from $X$ to $Y$
  • Reflexivity: Every map is homotopic to itself via the constant homotopy $H(x, t) = f(x)$
  • Symmetry: If $f \simeq g$, then $g \simeq f$ by reversing the homotopy $H(x, t) = H(x, 1-t)$
  • Transitivity: If $f \simeq g$ and $g \simeq h$, then $f \simeq h$ by concatenating the homotopies
• Homotopic maps induce the same homomorphisms on homotopy groups and homology groups
  • If $f \simeq g$, then $f_*: \pi_n(X) \to \pi_n(Y)$ and $g_*: \pi_n(X) \to \pi_n(Y)$ are equal for all $n \geq 0$
  • If $f \simeq g$, then $f_*: H_n(X) \to H_n(Y)$ and $g_*: H_n(X) \to H_n(Y)$ are equal for all $n \geq 0$

Homotopic Maps and Equivalences

Homotopic Maps

• Two continuous maps $f, g: X \to Y$ are homotopic if there exists a homotopy $H: X \times [0, 1] \to Y$ between them
  • Example: The identity map $id_X: X \to X$ and any constant map $c: X \to X$ are homotopic via $H(x, t) = (1-t)x + tc(x)$
• Homotopic maps share many topological properties, such as inducing the same homomorphisms on homotopy and homology groups

Homotopy Equivalences

• A homotopy equivalence between two topological spaces $X$ and $Y$ is a pair of continuous maps $f: X \to Y$ and $g: Y \to X$ such that $g \circ f \simeq id_X$ and $f \circ g \simeq id_Y$, where $id_X$ and $id_Y$ are the identity maps on $X$ and $Y$, respectively
  • The maps $f$ and $g$ are called homotopy equivalences or homotopy inverses of each other
• If a homotopy equivalence exists between $X$ and $Y$, then $X$ and $Y$ are said to be homotopy equivalent or have the same homotopy type, denoted by $X \simeq Y$
  • Example: The unit interval $[0, 1]$ and the unit circle $S^1$ are homotopy equivalent via the maps $f(t) = e^{2\pi it}$ and $g(z) = \frac{1}{2\pi i}\log z$
• Homotopy equivalent spaces share many topological properties, such as the same fundamental group, homology groups, and cohomology rings

Equivalence Relation of Homotopy

Reflexivity

• For any topological space $X$, the identity map $id_X: X \to X$ is a homotopy equivalence, as $id_X \circ id_X = id_X \simeq id_X$
  • The constant homotopy $H(x, t) = id_X(x)$ demonstrates that $id_X$ is homotopic to itself

Symmetry

• If $f: X \to Y$ is a homotopy equivalence with homotopy inverse $g: Y \to X$, then $g$ is also a homotopy equivalence with homotopy inverse $f$, as $f \circ g \simeq id_Y$ and $g \circ f \simeq id_X$
  • The homotopies demonstrating $f \circ g \simeq id_Y$ and $g \circ f \simeq id_X$ can be reversed to show $g \circ f \simeq id_X$ and $f \circ g \simeq id_Y$, respectively

Transitivity

• If $f: X \to Y$ and $g: Y \to Z$ are homotopy equivalences with homotopy inverses $f': Y \to X$ and $g': Z \to Y$, respectively, then $g \circ f: X \to Z$ is a homotopy equivalence with homotopy inverse $f' \circ g': Z \to X$
  • The composition of homotopies and the homotopy inverses demonstrate that $(g \circ f) \circ (f' \circ g') \simeq id_Z$ and $(f' \circ g') \circ (g \circ f) \simeq id_X$
  • Example: If $X \simeq Y$ and $Y \simeq Z$, then $X \simeq Z$ by composing the homotopy equivalences

Homotopy in Topological Classification

Classifying Spaces by Homotopy Type

• Homotopy provides a way to classify topological spaces by grouping together spaces that have the same homotopy type
  • Spaces that are homotopy equivalent share many important topological invariants, such as the fundamental group, homology groups, and cohomology rings
  • Example: All contractible spaces (spaces homotopy equivalent to a point) form a single homotopy equivalence class
• Homotopy equivalence is a weaker notion than homeomorphism, as homeomorphic spaces are always homotopy equivalent, but homotopy equivalent spaces may not be homeomorphic
  • Example: The punctured plane $\mathbb{R}^2 \setminus \{0\}$ and the unit circle $S^1$ are homotopy equivalent but not homeomorphic

Homotopy Groups and Distinguishing Spaces

• The study of homotopy groups, particularly the fundamental group, helps distinguish between spaces that are not homotopy equivalent and provides insight into the structure of topological spaces
  • The fundamental group $\pi_1(X)$ is a homotopy invariant that captures information about the loops in a space $X$
  • Higher homotopy groups $\pi_n(X)$ capture information about higher-dimensional spheres in $X$
  • Example: The fundamental group distinguishes between the torus $T^2$ and the sphere $S^2$, as $\pi_1(T^2) = \mathbb{Z} \times \mathbb{Z}$ and $\pi_1(S^2) = 0$
• Homotopy theory is a powerful tool in algebraic topology, allowing for the development of invariants and techniques to study and classify topological spaces