Simplicial Homology - Main Theorem

The fundamental theorem establishes simplicial homology as a well-defined topological invariant independent of triangulation choices.

Theorem

(Topological Invariance) If $K$ and $L$ are simplicial complexes with $|K| \cong |L|$ (homeomorphic geometric realizations), then $H_n(K) \cong H_n(L)$ for all $n \geq 0$ . Homology is a topological invariant.

This shows homology depends only on the underlying topological space, not on how we decompose it into simplices.

Theorem

(Homotopy Invariance) If $f, g : |K| \to |L|$ are homotopic continuous maps that are simplicial approximations, then the induced maps $f_*, g_* : H_n(K) \to H_n(L)$ are equal for all $n$ .

Proof sketch: Any continuous map can be approximated by a simplicial map after subdivision. Homotopic maps have chain-homotopic induced maps on chain complexes, which induce identical maps on homology.

Theorem

(Subdivision Invariance) If $K'$ is a subdivision of $K$ (each simplex of $K$ is subdivided into smaller simplices forming $K'$ ), then $H_n(K) \cong H_n(K')$ for all $n$ . The isomorphism is natural.

This allows us to compute homology with any convenient triangulation—finer subdivisions don't change the answer.

Theorem

(Mayer-Vietoris Sequence) If $K = K_1 \cup K_2$ where $K_1, K_2$ are subcomplexes, there is a long exact sequence: $\cdots \to H_n(K_1 \cap K_2) \to H_n(K_1) \oplus H_n(K_2) \to H_n(K) \to H_{n-1}(K_1 \cap K_2) \to \cdots$

This sequence is the homological analogue of Van Kampen's theorem, allowing computation of $H_n(K)$ from homology of pieces.

Example

For $S^n = D^n_+ \cup D^n_-$ (upper and lower hemispheres with overlap $S^{n-1} \times I \simeq S^{n-1}$ ), the Mayer-Vietoris sequence: $0 \to H_n(S^{n-1}) \to H_n(D^n_+) \oplus H_n(D^n_-) \to H_n(S^n) \to H_{n-1}(S^{n-1}) \to 0$

Since $H_n(D^n_\pm) = 0$ for $n > 0$ , we get $H_n(S^n) \cong H_{n-1}(S^{n-1})$ , yielding the sphere homology inductively.

Theorem

(Universal Coefficient Theorem) For any coefficient group $G$ , there is a split short exact sequence: $0 \to H_n(K) \otimes G \to H_n(K; G) \to \text{Tor}(H_{n-1}(K), G) \to 0$ relating homology with different coefficients.

This allows computing homology with coefficients in any abelian group from integer homology.