Gysin maps

Definition

Gysin maps are homomorphisms in algebraic topology that arise from the integration along the fiber of a proper map between two manifolds. These maps play a crucial role in relating the cohomology of the base space to the cohomology of the total space, facilitating push-forward operations in K-theory. They help in understanding how certain topological features behave under projection, particularly when examining fiber bundles and their associated classes.

5 Must Know Facts For Your Next Test

1. Gysin maps are typically denoted as $$g: H^*(X) \to H^{*+2}(Y)$$, where $X$ is the total space and $Y$ is the base space of a fibration.
2. They arise naturally in the context of smooth manifolds and allow for computations involving characteristic classes.
3. Gysin maps can be used to define push-forward operations that preserve important algebraic structures within cohomology.
4. The Gysin homomorphism is particularly useful when dealing with oriented manifolds and can be interpreted geometrically as integration over fibers.
5. In K-theory, Gysin maps help relate the K-theory groups of the base space and the total space, offering insights into how bundles behave under various transformations.

Review Questions

• How do Gysin maps facilitate the relationship between cohomology of fiber bundles and their base spaces?
  • Gysin maps establish a link between the cohomology of a fiber bundle's total space and that of its base space by allowing for an integration along fibers. This means that when you have a proper map from a total space to a base space, the Gysin map provides a systematic way to push forward cohomology classes from one space to another. The result is a richer understanding of how topological features transform when viewed through the lens of these fibers.
• Discuss the significance of Gysin maps in K-theory, especially in relation to push-forward operations.
  • In K-theory, Gysin maps play a crucial role by facilitating push-forward operations between different spaces. When studying vector bundles, these maps allow mathematicians to relate K-theory groups of the total space to those of the base space through a well-defined homomorphism. This connection is essential for understanding how vector bundles over base spaces behave under various topological transformations, thus providing valuable insights into algebraic topology.
• Evaluate how Gysin maps can be applied in computations involving characteristic classes and their implications on topology.
  • Gysin maps are instrumental in calculating characteristic classes by linking them with cohomology theories. By pushing forward cohomology classes via Gysin maps, one can derive important properties of bundles related to their curvature and other geometric attributes. These computations have far-reaching implications in topology, as they allow for a deeper understanding of manifold structures and their invariants, ultimately enhancing our ability to classify different topological spaces.

Related terms

Cohomology:

A mathematical tool used to study topological spaces through algebraic invariants, providing insights into their structure and properties.

Push-forward map:

A function that allows one to 'push' cohomology classes from one space to another, often related to integration over fibers in the context of Gysin maps.

Fiber bundle: A structure that consists of a space that looks locally like a product of two spaces, allowing for the study of sections and projections in topology.