🔢Elementary Algebraic Topology Unit 4 – Homotopy & the Fundamental Group

Key Concepts and Definitions

• Homotopy a continuous deformation of one path into another while keeping the endpoints fixed
• Path a continuous function $f: [0,1] \rightarrow X$ from the unit interval to a topological space $X$
• Loop a path that starts and ends at the same point, i.e., $f(0) = f(1)$
• Homotopy equivalence two spaces $X$ and $Y$ are homotopy equivalent if there exist continuous maps $f: X \rightarrow Y$ and $g: Y \rightarrow X$ such that $g \circ f \simeq id_X$ and $f \circ g \simeq id_Y$
  • $\simeq$ denotes homotopy equivalence
  • $id_X$ and $id_Y$ are the identity maps on $X$ and $Y$, respectively
• Fundamental group a group that captures the essential loop structure of a topological space
  • Denoted as $\pi_1(X, x_0)$, where $X$ is the space and $x_0$ is the basepoint
• Trivial fundamental group a fundamental group that consists only of the identity element (the constant loop)

Homotopy Basics

• Homotopy formalizes the idea of continuously deforming one path into another
• Two paths $f, g: [0,1] \rightarrow X$ are homotopic if there exists a continuous function $H: [0,1] \times [0,1] \rightarrow X$ such that:
  • $H(s, 0) = f(s)$ for all $s \in [0,1]$
  • $H(s, 1) = g(s)$ for all $s \in [0,1]$
  • $H(0, t) = f(0) = g(0)$ and $H(1, t) = f(1) = g(1)$ for all $t \in [0,1]$
• Homotopy defines an equivalence relation on the set of paths in a space
  • Reflexive: every path is homotopic to itself
  • Symmetric: if $f$ is homotopic to $g$, then $g$ is homotopic to $f$
  • Transitive: if $f$ is homotopic to $g$ and $g$ is homotopic to $h$, then $f$ is homotopic to $h$
• Homotopy classes the equivalence classes of paths under the homotopy relation
• Homotopy invariance topological properties that remain unchanged under homotopy equivalence (connectedness, dimension)

The Fundamental Group

• The fundamental group $\pi_1(X, x_0)$ is the set of homotopy classes of loops based at $x_0$ in the space $X$
• Group operation the operation in the fundamental group is concatenation of loops
  • Given two loops $f$ and $g$, their product $f * g$ is defined as:
    • $(f * g)(t) = f(2t)$ for $0 \leq t \leq 1/2$
    • $(f * g)(t) = g(2t - 1)$ for $1/2 \leq t \leq 1$
• Identity element the constant loop at the basepoint $x_0$
• Inverse element the reverse of a loop $f$, denoted as $f^{-1}$, where $f^{-1}(t) = f(1-t)$
• Fundamental group is a homotopy invariant if $X$ and $Y$ are homotopy equivalent, then $\pi_1(X, x_0) \cong \pi_1(Y, y_0)$ for any choice of basepoints $x_0$ and $y_0$
• Basepoint independence for path-connected spaces, the fundamental group is independent of the choice of basepoint up to isomorphism

Computing Fundamental Groups

• Trivial fundamental group for contractible spaces (spaces homotopy equivalent to a point), $\pi_1(X, x_0) = \{e\}$
• Fundamental group of the circle $\pi_1(S^1, x_0) \cong \mathbb{Z}$, generated by the homotopy class of the loop that goes around the circle once counterclockwise
• Fundamental group of a product if $X$ and $Y$ are path-connected, then $\pi_1(X \times Y, (x_0, y_0)) \cong \pi_1(X, x_0) \times \pi_1(Y, y_0)$
• Fundamental group of a wedge sum if $X$ and $Y$ are path-connected and $x_0 \in X$, $y_0 \in Y$ are the basepoints, then $\pi_1(X \vee Y, (x_0, y_0)) \cong \pi_1(X, x_0) * \pi_1(Y, y_0)$, where $*$ denotes the free product of groups
  • Wedge sum the space obtained by identifying the basepoints of two spaces (denoted by $\vee$)
• Van Kampen's theorem a tool for computing the fundamental group of a space that can be decomposed into simpler pieces with known fundamental groups
  • If $X = U \cup V$, where $U$ and $V$ are open, path-connected subspaces of $X$ and $U \cap V$ is also path-connected, then $\pi_1(X, x_0) \cong \pi_1(U, x_0) *_{\pi_1(U \cap V, x_0)} \pi_1(V, x_0)$, where $*_{\pi_1(U \cap V, x_0)}$ denotes the amalgamated free product

Applications and Examples

• Fundamental group of a torus $\pi_1(T^2) \cong \mathbb{Z} \times \mathbb{Z}$, generated by the homotopy classes of the two basic loops around the torus
• Fundamental group of a Klein bottle $\pi_1(K) \cong \langle a, b \mid aba^{-1}b \rangle$, a non-abelian group
• Fundamental group of a Möbius strip $\pi_1(M) \cong \mathbb{Z}$, generated by the homotopy class of the loop that goes around the strip once
• Fundamental group of a punctured plane $\pi_1(\mathbb{R}^2 \setminus \{0\}) \cong \mathbb{Z}$, generated by the homotopy class of a loop that goes around the puncture once
• Brouwer fixed point theorem if $f: D^2 \rightarrow D^2$ is a continuous map from the disk to itself, then $f$ has a fixed point
  • Proof uses the fact that $\pi_1(D^2) = \{e\}$ and $\pi_1(S^1) \cong \mathbb{Z}$
• Fundamental theorem of algebra every non-constant polynomial with complex coefficients has a root in $\mathbb{C}$
  • Proof uses the fact that $\pi_1(\mathbb{C} \setminus \{0\}) \cong \mathbb{Z}$

Relationship to Other Topological Concepts

• Covering spaces a covering space of $X$ is a space $\tilde{X}$ together with a map $p: \tilde{X} \rightarrow X$ such that every point in $X$ has an open neighborhood $U$ for which $p^{-1}(U)$ is a disjoint union of open sets in $\tilde{X}$, each mapped homeomorphically onto $U$ by $p$
  • Relationship between the fundamental groups of $X$ and $\tilde{X}$: $p_*(\pi_1(\tilde{X}, \tilde{x}_0))$ is a subgroup of $\pi_1(X, x_0)$, where $p(\tilde{x}_0) = x_0$
• Higher homotopy groups the $n$-th homotopy group $\pi_n(X, x_0)$ is the set of homotopy classes of maps from the $n$-dimensional sphere $S^n$ to $X$ that map a basepoint of $S^n$ to $x_0$
  • $\pi_0(X)$ the set of path components of $X$
  • $\pi_1(X, x_0)$ the fundamental group of $X$ based at $x_0$
  • $\pi_n(X, x_0)$ for $n \geq 2$ are abelian groups and are homotopy invariants
• Homology groups algebraic objects that capture the "holes" in a topological space
  • Relationship between the fundamental group and the first homology group: for a path-connected space $X$, there is a surjective homomorphism $\pi_1(X, x_0) \rightarrow H_1(X)$, called the Hurewicz homomorphism

Problem-Solving Techniques

• Identify the space and the basepoint when computing the fundamental group
• Determine if the space is path-connected if not, consider each path component separately
• Look for ways to decompose the space into simpler pieces (e.g., using the wedge sum or Van Kampen's theorem)
• Consider the space's relationship to known spaces (e.g., contractible spaces, circles, products)
• Use the properties of the fundamental group (e.g., homotopy invariance, basepoint independence) to simplify the problem
• Visualize loops and deformations in the space to build intuition
• Consider the relationships between the fundamental group and other topological invariants (e.g., covering spaces, higher homotopy groups, homology groups)
• Practice computing fundamental groups for a variety of spaces to develop problem-solving skills

Further Reading and Resources

• "Algebraic Topology" by Allen Hatcher a comprehensive textbook covering the fundamental group and related topics
• "Topology" by James Munkres another popular textbook with a chapter dedicated to the fundamental group
• "Fundamental Groups and Covering Spaces" by Elon Lages Lima a concise introduction to the fundamental group and its relationship to covering spaces
• "Algebraic Topology: A First Course" by William Fulton a more advanced textbook with a focus on algebraic methods
• "Topology and Groupoids" by Ronald Brown an alternative approach to algebraic topology using groupoids, with a chapter on the fundamental groupoid
• "Homotopy Theory: An Introduction to Algebraic Topology" by Brayton Gray a textbook that develops algebraic topology from the perspective of homotopy theory
• Online resources:
  • "Algebraic Topology" by Richard Elman, Nikolai Karpenko, and Alexander Merkurjev (lecture notes)
  • "Fundamental Group" by Aisling McCluskey and Brian McMaster (Wolfram MathWorld article)
  • "The Fundamental Group" by Keith Conrad (expository paper)