Singular Homology - Applications

Singular homology's generality and computational power make it indispensable across mathematics, from topology to algebraic geometry and data science.

Theorem

(Lefschetz Fixed Point Theorem) Let $f : X \to X$ be a continuous map of a compact triangulable space. The Lefschetz number is: $\Lambda(f) = \sum_{n=0}^\infty (-1)^n \text{tr}(f_* : H_n(X; \mathbb{Q}) \to H_n(X; \mathbb{Q}))$ If $\Lambda(f) \neq 0$ , then $f$ has a fixed point.

This generalizes Brouwer: for $f : D^n \to D^n$ , we have $\Lambda(f) = 1 \neq 0$ , forcing a fixed point.

Example

No retraction from ball to sphere: If $r : D^n \to S^{n-1}$ were a retraction, then $r_* : H_{n-1}(D^n) = 0 \to H_{n-1}(S^{n-1}) = \mathbb{Z}$ composed with inclusion would be identity on $\mathbb{Z}$ , impossible.

Theorem

(Degree Theory) For a map $f : S^n \to S^n$ , the induced map $f_* : H_n(S^n) \to H_n(S^n)$ is multiplication by an integer $\deg(f)$ , the degree of $f$ . Properties:

$\deg(\text{id}) = 1$ , $\deg(\text{const}) = 0$
$\deg(g \circ f) = \deg(g) \cdot \deg(f)$
Homotopic maps have equal degree
Antipodal map has degree $(-1)^{n+1}$

Example

Fundamental theorem of algebra: Every non-constant polynomial $p : \mathbb{C} \to \mathbb{C}$ has a root. Proof: For large $|z|$ , $p(z)/z^n$ is close to the leading coefficient. This defines a homotopy from $z \mapsto z^n$ to $z \mapsto p(z)/|p(z)|$ on large circles, giving $\deg(p|_{S^1}) = n \neq 0$ . If $p$ had no roots, it would extend to a map $D^2 \to S^1$ , contradicting $\pi_1(S^1) = \mathbb{Z} \neq 0 = \pi_1(D^2)$ .

Theorem

(Homological Algebra) Singular homology initiated homological algebra: the study of chain complexes, exact sequences, derived functors (Ext, Tor), spectral sequences, and category theory. These tools pervade modern mathematics.

Example

Topological data analysis (TDA): Persistent homology tracks how $H_n$ changes across a filtration of spaces built from data points. Applications include:

Shape analysis in computer vision
Neuroscience (brain networks)
Material science (porous media)
Time series analysis

Remark

Singular homology extends to sheaf cohomology in algebraic geometry, computing cohomology of coherent sheaves on schemes. This connects topology with number theory and arithmetic geometry, enabling proofs of the Weil conjectures and more.

Theorem

(Hurewicz and Whitehead) Homology and homotopy groups connect deeply:

$H_1(X) \cong \pi_1(X)/[\pi_1, \pi_1]$ (abelianization)
Hurewicz theorem: first non-vanishing homotopy equals first non-vanishing homology
Whitehead: map inducing isomorphisms on all $H_n$ induces isomorphisms on all $\pi_n$ for simply-connected CW complexes