Algebraic Topology: Functors and Invariants

Algebraic topology provides a bridge between the continuous world of topology and the discrete, structured world of algebra. The central goal is to assign algebraic invariants (groups, rings, etc.) to topological spaces such that homeomorphic—and often homotopy equivalent—spaces possess isomorphic algebraic structures.

1. Homotopy and Homotopy Equivalence

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

A map $f:X\to Y$ is a homotopy equivalence if there exists a map $g:Y\to X$ such that $g\circ f\simeq {\text{id}}_X$ and $f\circ g\simeq {\text{id}}_Y$. If such a map exists, we say $X$ and $Y$ have the same homotopy type ($X\simeq Y$).

Homotopy equivalence is a coarser relation than homeomorphism; for instance, ${\mathbb{R}}^n$ is homotopy equivalent to a point (it is contractible), though they are clearly not homeomorphic for $n>0$.

2. The Fundamental Group ${\pi }_1(X,x_0)$

The fundamental group captures the way loops can be drawn in a space. Let $X$ be a topological space and $x_0\in X$ a base point. A loop is a map $\gamma :[0,1]\to X$ with $\gamma (0)=\gamma (1)=x_0$.

We define ${\pi }_1(X,x_0)$ as the set of homotopy classes of loops (relative to endpoints), where the group operation is the concatenation of loops: $[\gamma ]\cdot [\sigma ]=\{\begin{array}{ll}\gamma (2t)& 0\le t\le 1/2\\ \sigma (2t-1)& 1/2\le t\le 1\end{array}$

The Fundamental Group of the Circle

A classic result is ${\pi }_1(S^1)\cong \mathbb{Z}$. This is proved by considering the universal cover $p:\mathbb{R}\to S^1$ given by $t\mapsto e^{2\pi it}$. A loop in $S^1$ lifts to a path in $\mathbb{R}$ starting at $0$ and ending at some $n\in \mathbb{Z}$, which is the degree (or winding number) of the loop.

3. Covering Spaces

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

The Path Lifting Property and Homotopy Lifting Property are fundamental:

1. Given a path $f:I\to X$ and a lift ${\tilde{x}}_0$ of $f(0)$, there is a unique lift $\tilde{f}:I\to \tilde{X}$.
2. A homotopy $F:Y\times I\to X$ lifts uniquely given a lift of $F{|}_{Y\times \{0\}}$.

These properties imply that $p_{\ast }:{\pi }_1(\tilde{X},{\tilde{x}}_0)\to {\pi }_1(X,x_0)$ is an injective homomorphism, and its image is a subgroup whose conjugacy class corresponds to the covering $\tilde{X}$.

4. Simplicial Homology

While ${\pi }_1$ captures 1-dimensional “holes,” homology provides a way to detect $n$-dimensional holes.

An $n$-simplex ${\Delta }^n$ is the convex hull of $n+1$ points in general position. A simplicial complex $K$ is a collection of simplices such that every face of a simplex in $K$ is also in $K$.

Let $C_n(X)$ be the free abelian group generated by the $n$-simplices of $X$. Elements of $C_n(X)$ are called $n$-chains.

The Boundary Operator

The boundary operator ${\partial }_n:C_n(X)\to C_{n-1}(X)$ is defined linearly on simplices by: ${\partial }_n(v_0,\dots ,v_n)={\sum }_{i=0}^n(-1{)}^i(v_0,\dots ,{\hat{v}}_i,\dots ,v_n)$ where ${\hat{v}}_i$ denotes that the vertex $v_i$ is omitted.

Lemma: ${\partial }_n\circ {\partial }_{n+1}=0$. Proof: $\partial \partial (v_0,\dots ,v_{n+1})={\sum }_{j<i}(-1{)}^i(-1{)}^j(v_0,\dots ,{\hat{v}}_j,\dots ,{\hat{v}}_i,\dots ,v_{n+1})+{\sum }_{j>i}(-1{)}^i(-1{)}^{j-1}(v_0,\dots ,{\hat{v}}_i,\dots ,{\hat{v}}_j,\dots ,v_{n+1})$ Each term $(v_0,\dots ,{\hat{v}}_j,\dots ,{\hat{v}}_i,\dots ,v_{n+1})$ appears twice with opposite signs (due to the $j<i$ vs $j>i$ indexing), so they cancel.

Because $\text{im }{\partial }_{n+1}\subset \ker {\partial }_n$, we define the $n$-th homology group: $H_n(X)=\ker {\partial }_n/\text{im }{\partial }_{n+1}$

5. Betti Numbers and Euler Characteristic

The $n$-th Betti number $b_n$ is the rank of the free part of $H_n(X)$. Intuitively:

• $b_0$ is the number of connected components.
• $b_1$ is the number of 1-dimensional holes (loops).
• $b_2$ is the number of 2-dimensional holes (voids).

The Euler Characteristic $\chi$ is a topological invariant defined as: $\chi (X)={\sum }_{n=0}^{\infty }(-1{)}^n\text{rank}(H_n(X))$ For a finite simplicial complex with $V$ vertices, $E$ edges, and $F$ faces, this identity holds: $\chi =V-E+F$.

6. Functoriality

Algebraic topology is fundamentally functorial. A continuous map $f:X\to Y$ induces:

1. A homomorphism $f_{\ast }:{\pi }_1(X,x_0)\to {\pi }_1(Y,f(x_0))$.
2. Homomorphisms $f_{\ast }:H_n(X)\to H_n(Y)$ for all $n$.

Crucially, $(g\circ f{)}_{\ast }=g_{\ast }\circ f_{\ast }$ and $({\text{id}}_X{)}_{\ast }={\text{id}}_{H(X)}$. This allows us to use algebra to prove topological impossibilities.

7. Famous Applications

Brouwer Fixed Point Theorem

Theorem: Every continuous map $f:D^n\to D^n$ has a fixed point. Proof: If $f$ has no fixed point, we can construct a retraction $r:D^n\to S^{n-1}$ by sending $x$ to the intersection of the ray from $f(x)$ through $x$ with the boundary $S^{n-1}$. However, $H_{n-1}(S^{n-1})\cong \mathbb{Z}$ and $H_{n-1}(D^n)=0$. Functoriality would require the identity on $\mathbb{Z}$ to factor through 0, a contradiction.

The Hairy Ball Theorem

Theorem: A continuous tangent vector field on $S^n$ must vanish at some point if $n$ is even. This follows from the fact that if a non-vanishing vector field existed, the antipodal map would be homotopic to the identity. However, the antipodal map has degree $(-1{)}^{n+1}$, while the identity has degree 1. For even $n$, these are different.

Computational Example: Euler Characteristic

The following Python snippet computes the Euler characteristic of a simplicial complex represented by its simplices across dimensions.

def compute_euler_characteristic(simplices_by_dim):
    """
    simplices_by_dim: Dict where keys are dimension (int) 
    and values are lists of simplices (tuples of vertex indices).
    """
    chi = 0
    for dim, simplices in simplices_by_dim.items():
        count = len(simplices)
        # Standard alternating sum Formula: sum (-1)^n * n_k
        if dim % 2 == 0:
            chi += count
        else:
            chi -= count
    return chi

# Example: A Tetrahedron (homeomorphic to S^2)
# 4 vertices, 6 edges, 4 faces
tetrahedron = {
    0: [(0,), (1,), (2,), (3,)],
    1: [(0,1), (0,2), (0,3), (1,2), (1,3), (2,3)],
    2: [(0,1,2), (0,1,3), (0,2,3), (1,2,3)]
}

chi_sphere = compute_euler_characteristic(tetrahedron)
print(f"Euler Characteristic of Sphere (Tetrahedron): {chi_sphere}")

# Example: A Torus (triangulated)
# A minimal triangulation of the torus requires V=7, E=21, F=14.
torus = {
    0: [i for i in range(7)],
    1: [i for i in range(21)],
    2: [i for i in range(14)]
}
chi_torus = compute_euler_characteristic(torus)
print(f"Euler Characteristic of Torus: {chi_torus}")

Advanced Quiz