7.4 Applications in algebra and topology

The Tor and Ext functors play crucial roles in algebra and topology. They help us understand relationships between homology and cohomology, compute homology of product spaces, and classify group extensions.

These functors appear in key theorems like the Universal Coefficient Theorem and Künneth formula. They also show up in cohomology ring structures and spectral sequences, providing powerful tools for analyzing topological spaces.

Universal Coefficient and Künneth Theorems

Relationship between Homology and Cohomology

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• Universal coefficient theorem establishes a relationship between homology and cohomology groups
  • States that for a chain complex $C$ of free abelian groups, there is a short exact sequence: $0 \to Ext(H_{n-1}(C), G) \to H^n(C;G) \to Hom(H_n(C), G) \to 0$
  • Allows computation of cohomology groups from homology groups and vice versa (singular homology, cellular homology)
• Künneth formula computes the homology groups of a tensor product of two chain complexes
  • For chain complexes $C$ and $D$, and a principal ideal domain $R$, there is a short exact sequence: $0 \to \bigoplus_{i+j=n} H_i(C) \otimes_R H_j(D) \to H_n(C \otimes_R D) \to \bigoplus_{i+j=n-1} Tor_1^R(H_i(C), H_j(D)) \to 0$
  • Allows computation of homology groups of product spaces (torus, Klein bottle)

Algebraic Interpretations of Ext and Tor

• Ext functor can be interpreted as classifying group extensions
  • For abelian groups $A$ and $B$, $Ext(A,B)$ classifies extensions of the form: $0 \to B \to E \to A \to 0$
  • Elements of $Ext(A,B)$ correspond to equivalence classes of such extensions (central extensions, split extensions)
• Tor functor appears in the homology of groups
  • For a group $G$ and a $G$-module $M$, there is a long exact sequence: $\dots \to H_n(G;M) \to H_n(G) \otimes M \to Tor_1^{\mathbb{Z}G}(H_{n-1}(G), M) \to H_{n-1}(G;M) \to \dots$
  • Tor measures the failure of the homology of $G$ to be a flat module over the group ring $\mathbb{Z}G$ (group homology, Lyndon-Hochschild-Serre spectral sequence)

Cohomology and Spectral Sequences

Cohomology Ring Structure

• Cohomology groups $H^n(X;R)$ of a space $X$ with coefficients in a ring $R$ form a graded ring
  • The product is given by the cup product: $\smile: H^i(X;R) \otimes H^j(X;R) \to H^{i+j}(X;R)$
  • The cup product is induced by the diagonal map $X \to X \times X$ (Poincaré duality, Künneth formula for cohomology)
• The cohomology ring encodes important topological information about the space
  • For example, the cohomology ring of a compact oriented manifold determines its homotopy type (rational homotopy theory)

Spectral Sequences and Local Cohomology

• Eilenberg-Moore spectral sequence relates the cohomology of a fiber bundle to the cohomology of its base and fiber
  • For a fibration $F \to E \to B$ with $B$ simply connected, there is a spectral sequence: $E_2^{p,q} = Tor_{p,q}^{H^*(B)}(H^*(E), \mathbb{Z}) \Rightarrow H^{p+q}(F)$
  • The spectral sequence converges to the cohomology of the fiber $F$ (Serre spectral sequence, Leray-Serre spectral sequence)
• Local cohomology is a cohomology theory that captures local properties of a space near a subspace
  • For a space $X$ and a subspace $Y$, the local cohomology groups $H^i_Y(X;R)$ fit into a long exact sequence: $\dots \to H^i_Y(X;R) \to H^i(X;R) \to H^i(X-Y;R) \to H^{i+1}_Y(X;R) \to \dots$
  • Local cohomology is related to sheaf cohomology and has applications in algebraic geometry (Grothendieck's local cohomology theory)