Cohomology of Spheres

Definition

Cohomology of spheres refers to the study of cohomological properties of sphere spaces, particularly their cohomology groups. These groups, denoted as $H^n(S^k; R)$, describe the algebraic structure and topological features of spheres in terms of ring theory and module theory over a coefficient ring $R$. This area connects deeply with concepts like the excision theorem and Alexandrov-Čech cohomology, which provide tools for analyzing how these groups behave under certain conditions and mappings.

5 Must Know Facts For Your Next Test

1. The cohomology groups of spheres have been determined for all dimensions, with the key results being $H^n(S^k; R) = 0$ for $n < k$ and $H^k(S^k; R) \cong R$.
2. Cohomology of spheres is crucial in understanding the properties of manifolds, as many results about manifolds can be deduced from the behavior of spheres.
3. The cohomology ring structure for spheres reveals a rich algebraic aspect, often leading to the use of cup products to study interactions between different degrees.
4. In cases where the coefficient ring $R$ is a field, the cohomology groups are particularly clean and easier to analyze compared to general coefficients.
5. The concepts introduced by excision in relation to the cohomology of spheres allow mathematicians to simplify complicated spaces into manageable pieces while retaining their topological properties.

Review Questions

• How do the cohomology groups of spheres illustrate the application of the excision theorem?
  • The cohomology groups of spheres demonstrate the excision theorem by allowing us to consider smaller subspaces within larger spheres. When we remove certain 'nice' subsets from a sphere, the excision theorem assures that the cohomological properties remain unchanged. This facilitates studying complex shapes by focusing on simpler parts without losing important topological information.
• Discuss how Alexandrov-Čech cohomology can be applied to calculate the cohomology groups of spheres.
  • Alexandrov-Čech cohomology provides a method for calculating cohomology groups using covers of spaces, making it particularly useful for spheres. By using an open cover consisting of basic neighborhoods around points on the sphere, we can derive Čech cohomology groups that coincide with singular cohomology groups in this case. This approach not only confirms known results but also offers insights into more general topological spaces.
• Evaluate the significance of the structure of cohomology rings in understanding the interactions between different degrees in the context of spheres.
  • The structure of cohomology rings plays a pivotal role in understanding how different degrees interact through operations like cup products. For spheres, this structure reveals symmetries and relationships among various dimensions that can provide insights into manifold topology and homotopy theory. By analyzing these interactions within cohomology rings, mathematicians can infer broader results about classifying spaces and their respective topological features.

Related terms

Cohomology Groups: Algebraic structures that assign a sequence of abelian groups or modules to a topological space, reflecting its shape and features.

Excision Theorem: A fundamental result in algebraic topology that allows certain types of 'removal' or 'cutting' in spaces without changing their cohomological properties.

Simplicial Complex: A type of topological space constructed by gluing together points, line segments, triangles, and their higher-dimensional counterparts.