13.3 Computation of cohomology groups for simple manifolds

De Rham cohomology groups offer powerful tools for understanding manifolds' topological properties. This section dives into calculating these groups for simple yet important spaces like spheres, tori, and projective spaces.

We'll explore how cohomology reveals key features of these manifolds, including their Betti numbers and Euler characteristics. These calculations showcase the practical application of cohomology theory in topology.

Cohomology of Simple Manifolds

Sphere Cohomology and Torus Cohomology

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• Sphere cohomology calculates the de Rham cohomology groups of n-dimensional spheres
• For an n-sphere, cohomology groups consist of $H^0(S^n) \cong \mathbb{R}$ and $H^n(S^n) \cong \mathbb{R}$
• All other cohomology groups of spheres vanish $H^k(S^n) \cong 0$ for $0 < k < n$
• Torus cohomology examines the de Rham cohomology groups of n-dimensional tori
• For an n-torus, cohomology groups follow the pattern $H^k(T^n) \cong \mathbb{R}^{\binom{n}{k}}$
• Betti numbers of tori correspond to binomial coefficients (Pascal's triangle)
• Künneth formula applies to torus cohomology due to its product structure

Projective Space Cohomology and Euler Characteristic

• Projective space cohomology studies de Rham cohomology groups of real and complex projective spaces
• For real projective space $\mathbb{RP}^n$, cohomology groups alternate between $\mathbb{R}$ and 0
• Complex projective space $\mathbb{CP}^n$ cohomology groups follow the pattern $H^{2k}(\mathbb{CP}^n) \cong \mathbb{R}$ for $0 \leq k \leq n$
• Euler characteristic relates to alternating sum of Betti numbers $\chi(M) = \sum_{k=0}^n (-1)^k b_k$
• Euler characteristic remains invariant under continuous deformations (homotopy invariance)
• Provides topological information about manifolds (surfaces with handles, Klein bottle)

Cohomological Operations and Duality

Cup Product and Künneth Formula

• Cup product defines multiplication on cohomology classes $H^k(M) \times H^l(M) \to H^{k+l}(M)$
• Graded-commutative operation satisfies $\alpha \cup \beta = (-1)^{kl} \beta \cup \alpha$ for $\alpha \in H^k(M)$ and $\beta \in H^l(M)$
• Induces ring structure on cohomology, enhancing algebraic properties
• Künneth formula computes cohomology of product spaces $H^*(X \times Y) \cong H^*(X) \otimes H^*(Y)$
• Allows decomposition of cohomology groups for product manifolds
• Applies to torus cohomology calculations $H^*(T^n) \cong H^*(S^1)^{\otimes n}$

Poincaré Duality and Hodge Theory

• Poincaré duality establishes isomorphism between cohomology groups of complementary dimensions
• For compact oriented n-manifold M, $H^k(M) \cong H^{n-k}(M)$ for all k
• Relates homology and cohomology groups via $H_k(M) \cong H^{n-k}(M)$
• Hodge theory connects de Rham cohomology to harmonic forms on Riemannian manifolds
• Decomposes k-forms into exact, coexact, and harmonic components (Hodge decomposition)
• Establishes isomorphism between cohomology classes and harmonic forms
• Hodge star operator $\star$ plays crucial role in relating forms of complementary degrees