Glossary

13.3 Computation of cohomology groups for simple manifolds

2 min read•august 9, 2024

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 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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  • cohomology calculates the de Rham cohomology groups of n-dimensional spheres
  • For an n-sphere, cohomology groups consist of H0(Sn)≅RH^0(S^n) \cong \mathbb{R} and Hn(Sn)≅RH^n(S^n) \cong \mathbb{R}
  • All other cohomology groups of spheres vanish Hk(Sn)≅0H^k(S^n) \cong 0 for 0<k<n0 < k < n
  • cohomology examines the de Rham cohomology groups of n-dimensional tori
  • For an n-torus, cohomology groups follow the pattern Hk(Tn)≅R(nk)H^k(T^n) \cong \mathbb{R}^{\binom{n}{k}}
  • Betti numbers of tori correspond to binomial coefficients (Pascal's triangle)
  • 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 RPn\mathbb{RP}^n, cohomology groups alternate between R\mathbb{R} and 0
  • Complex projective space CPn\mathbb{CP}^n cohomology groups follow the pattern H2k(CPn)≅RH^{2k}(\mathbb{CP}^n) \cong \mathbb{R} for 0≤k≤n0 \leq k \leq n
  • relates to alternating sum of Betti numbers χ(M)=∑k=0n(−1)kbk\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 Hk(M)×Hl(M)→Hk+l(M)H^k(M) \times H^l(M) \to H^{k+l}(M)
  • Graded-commutative operation satisfies α∪β=(−1)klβ∪α\alpha \cup \beta = (-1)^{kl} \beta \cup \alpha for α∈Hk(M)\alpha \in H^k(M) and β∈Hl(M)\beta \in H^l(M)
  • Induces ring structure on cohomology, enhancing algebraic properties
  • Künneth formula computes cohomology of product spaces H∗(X×Y)≅H∗(X)⊗H∗(Y)H^*(X \times Y) \cong H^*(X) \otimes H^*(Y)
  • Allows decomposition of cohomology groups for product manifolds
  • Applies to torus cohomology calculations H∗(Tn)≅H∗(S1)⊗nH^*(T^n) \cong H^*(S^1)^{\otimes n}

Poincaré Duality and Hodge Theory

  • establishes between cohomology groups of complementary dimensions
  • For compact oriented n-manifold M, Hk(M)≅Hn−k(M)H^k(M) \cong H^{n-k}(M) for all k
  • Relates homology and cohomology groups via Hk(M)≅Hn−k(M)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

Key Terms to Review (20)

Betti Numbers: Betti numbers are topological invariants that provide a way to measure the number of 'holes' in a topological space at different dimensions. They are crucial in understanding the shape and structure of manifolds, as they relate to the dimensions of cohomology groups, which can be computed through various methods, including De Rham cohomology.
čech cohomology: Čech cohomology is a tool in algebraic topology that associates a sequence of abelian groups or vector spaces to a topological space, providing a way to study its global properties through local data. It is particularly useful for computing cohomology groups of spaces that may not be well-behaved in a classical sense, especially for simple manifolds. By using open covers and taking limits, it captures the essential topological features of spaces that can be analyzed via continuous maps and sheaf theory.
Cell complex: A cell complex is a type of topological space constructed by gluing together disks of various dimensions. It serves as a foundational structure in algebraic topology, allowing for the analysis and classification of spaces based on their cellular structure. Cell complexes can be used to calculate homology and cohomology groups, which are essential tools for understanding the topological properties of spaces.
Characteristic Classes: Characteristic classes are a way to associate topological invariants to vector bundles, providing important information about their geometry and topology. They serve as a powerful tool in differential topology, allowing for the classification of bundles and the understanding of various manifold properties through their cohomology. Characteristic classes connect with essential concepts like partitions of unity and cohomology groups, while also playing a role in applications related to fixed point theory.
Coboundary: A coboundary is a specific type of cochain that represents the image of a boundary operator acting on a chain in a cohomology theory. It connects to other concepts by demonstrating how cochains can be related to the topology of manifolds, especially in computing their cohomology groups, which provides crucial information about the manifold's structure and properties.
Cocycle: A cocycle is a function that assigns values to the intersections of open sets in a cover of a topological space, satisfying certain conditions related to the structure of the space. This concept is crucial in cohomology, where cocycles are used to define cohomology classes and play an essential role in the computation of cohomology groups for manifolds. Understanding cocycles helps in analyzing how these classes can vary and interact under continuous transformations.
De Rham Cohomology: De Rham cohomology is a mathematical tool used in differential geometry and topology that studies the global properties of smooth manifolds through differential forms and their equivalence classes. It provides a bridge between analysis and topology by associating differential forms with topological invariants, allowing for deeper insights into the structure of manifolds. This approach is particularly useful when combined with concepts like partitions of unity, exterior algebra, and cohomology groups.
Euler characteristic: The Euler characteristic is a topological invariant that provides a way to classify surfaces and other geometric objects based on their shape and structure. It is calculated using the formula $$ ext{Euler characteristic} = V - E + F$$, where V represents the number of vertices, E represents the number of edges, and F represents the number of faces in a polyhedron. This concept connects deeply with various fields, including differential topology, by offering insights into the properties and classifications of manifolds, cohomology groups, and mapping degrees.
Fibration: A fibration is a structure in topology that allows for a way to systematically relate spaces through continuous mappings, typically characterized by having a homotopy lifting property. This concept connects various topological spaces and plays a vital role in understanding their cohomological properties, particularly in the computation of cohomology groups for simple manifolds.
Henri Poincaré: Henri Poincaré was a French mathematician and physicist, often regarded as one of the founders of topology and a pioneer in the study of dynamical systems. His work laid foundational concepts that connect with various branches of mathematics, especially in understanding the behavior of continuous functions and spaces.
Isomorphism: Isomorphism refers to a mathematical structure-preserving mapping between two objects, indicating that they are essentially the same in terms of their algebraic or topological properties. This concept emphasizes the idea that even if two structures may look different, they can still behave identically under certain operations or transformations. Understanding isomorphisms is crucial for comparing and classifying different structures in various mathematical fields.
John Milnor: John Milnor is a prominent American mathematician known for his significant contributions to differential topology, particularly in the areas of manifold theory, Morse theory, and the topology of high-dimensional spaces. His work has fundamentally shaped the field and has broad implications for various topics within topology, including submersions, critical values, and cohomology groups.
Künneth Formula: The Künneth formula is a powerful tool in algebraic topology that describes how the cohomology groups of the product of two topological spaces relate to the cohomology groups of the individual spaces. This formula provides a way to compute the cohomology of a product space by connecting the cohomological properties of its factors, often yielding new insights into their topological structure.
Leray Spectral Sequence: The Leray spectral sequence is a tool in algebraic topology that provides a way to compute the cohomology groups of a space from the cohomology of its fibers and base space, particularly in the context of fibrations. It connects the cohomological properties of a fibration to those of its total space and base, which makes it especially useful for computing cohomology groups for simple manifolds. The spectral sequence is constructed from the derived functors of sheaf cohomology and reveals deep relationships between the topological features of the space involved.
Mayer-Vietoris sequence: The Mayer-Vietoris sequence is a powerful tool in algebraic topology that provides a way to compute the homology and cohomology groups of a topological space by breaking it down into simpler pieces. This sequence arises when a topological space can be decomposed into two open sets whose intersection is also open, allowing for a systematic way to relate the homology groups of the individual pieces to the whole space. It serves as a bridge connecting local properties of spaces to global topological features.
Poincaré Duality: Poincaré Duality is a fundamental theorem in algebraic topology that establishes a deep relationship between the homology and cohomology groups of a manifold. It asserts that, for a closed orientable manifold of dimension n, the k-th homology group is isomorphic to the (n-k)-th cohomology group. This duality highlights the interplay between these two topological invariants and provides crucial insights into the structure of manifolds.
Singular Cohomology: Singular cohomology is a mathematical tool used in algebraic topology to assign algebraic invariants, called cohomology groups, to topological spaces. It helps in understanding the structure of spaces by encoding information about their shape and connectivity through singular chains, which are formal sums of continuous maps from standard simplices into the space. By analyzing these chains, singular cohomology provides powerful insights into the properties of spaces, particularly simple manifolds, where it is instrumental in computing cohomology groups.
Sphere: A sphere is a perfectly symmetrical three-dimensional shape where every point on its surface is equidistant from its center. In topology, spheres serve as fundamental examples of manifolds, helping illustrate complex structures like tori and projective spaces, and play a crucial role in understanding smooth structures and cohomology groups.
Torus: A torus is a surface shaped like a doughnut, characterized by a hole in the center and formed by revolving a circle around an axis that does not intersect the circle. This unique geometric structure serves as a fundamental example of a manifold, illustrating key concepts like product manifolds and quotient manifolds, as well as offering insights into cohomology groups and homology in algebraic topology.
Triviality: In mathematics, triviality refers to the simplest or most basic cases of a mathematical object or structure, often devoid of interesting properties. It indicates a lack of complexity or richness in the context of cohomology groups, particularly when assessing the nature of various manifolds. This concept is essential for understanding how cohomology groups can reveal intricate topological features of manifolds, while trivial cases may lead to limited insights.
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