Homotopy pushout

Definition

A homotopy pushout is a construction in algebraic topology that combines spaces along a shared subspace to create a new space that reflects the 'homotopical' nature of the original spaces. This process allows us to understand how maps between spaces behave under continuous deformation, which is essential when studying homotopy equivalences. It provides a way to glue spaces together while preserving their homotopical properties, making it a powerful tool in the analysis of topological spaces.

5 Must Know Facts For Your Next Test

1. Homotopy pushouts are used to analyze how spaces behave when gluing along a common subspace, especially in the context of deformation retracts.
2. The construction of a homotopy pushout depends on the choice of maps between the spaces involved, which are crucial for determining the resulting topology.
3. In many cases, homotopy pushouts can be shown to preserve certain properties like connectedness and local contractibility of the involved spaces.
4. They play an important role in understanding functors in homotopy theory, particularly when studying derived functors and model categories.
5. Homotopy pushouts can be computed using diagrams in category theory, providing a bridge between algebraic topology and abstract category theory.

Review Questions

• How does the concept of homotopy pushout relate to the idea of gluing spaces together?
  • Homotopy pushouts serve as a method for gluing spaces along shared subspaces while maintaining their homotopical characteristics. When two or more spaces are mapped into a common subspace, the homotopy pushout constructs a new space that reflects the relationships between the original spaces through continuous deformations. This process helps us understand how these glued spaces interact and retain their essential topological features.
• Discuss the importance of maps in determining the properties of a homotopy pushout.
  • The maps used in constructing a homotopy pushout are critical because they define how the original spaces interact with each other and how they glue together at the shared subspace. The choice of these maps influences key properties such as connectedness, local contractibility, and other topological features of the resulting space. If the maps are not carefully selected, the resulting homotopy pushout may not preserve desired properties or could lead to unexpected topological configurations.
• Evaluate the implications of homotopy pushouts on our understanding of functors within algebraic topology.
  • Homotopy pushouts have significant implications for our understanding of functors in algebraic topology, particularly when discussing derived functors and model categories. They allow us to construct new spaces from existing ones while preserving essential homotopical properties. This process highlights how algebraic topology can leverage categorical concepts to create functorial relationships, enriching our knowledge about topological structures and their interactions. Such insights contribute to advances in both theoretical and applied mathematics.

Related terms

Pushout: A pushout is a categorical concept where two spaces are combined over a common subspace, resulting in a new space that identifies points according to specified maps.

Homotopy equivalence: A homotopy equivalence is a relationship between two topological spaces indicating that they can be transformed into each other through continuous deformations.

Mapping cone: The mapping cone is a construction used in topology to study homotopies, formed by taking a space and attaching a cone based on a continuous map.