Mapping cone

Definition

A mapping cone is a construction in algebraic topology that associates to a continuous map between topological spaces a new space that intuitively represents the 'cylinder' of the map. This concept helps in understanding how spaces change under continuous transformations and plays a crucial role in analyzing homotopy equivalences and fiber sequences, providing a way to study the topological properties of spaces when maps are involved.

5 Must Know Facts For Your Next Test

1. The mapping cone can be thought of as the quotient space obtained by attaching a cone to one end of the domain space through the map.
2. It can be used to analyze homotopy types and is instrumental in constructing long exact sequences in homology.
3. The mapping cone retains information about the original space and the nature of the map, allowing for insights into their topological properties.
4. In the context of homotopy fiber sequences, mapping cones help show how spaces relate to each other via maps and illustrate how fibers behave under deformation.
5. The mapping cone is a key tool for establishing results like the excision theorem in homology, revealing connections between different spaces through continuous mappings.

Review Questions

• How does the construction of a mapping cone reflect the properties of the original space and the continuous map involved?
  • The construction of a mapping cone reflects the properties of both the original space and the continuous map by capturing how the space is altered when attached to the cone. Essentially, when you create a mapping cone from a space via a specific map, you’re adding new 'directions' based on how points in the original space are related to points in the cone. This highlights crucial changes that happen due to continuous transformations, which can influence overall homotopy characteristics.
• Discuss how mapping cones contribute to understanding homotopy equivalence in topological spaces.
  • Mapping cones play an important role in understanding homotopy equivalence by providing a framework through which we can visualize and analyze maps between spaces. When two spaces are homotopy equivalent, their mapping cones will also share certain topological features that reflect this relationship. By studying these cones, one can derive insights about how spaces relate through maps and how they maintain their essential properties despite transformations.
• Evaluate the significance of mapping cones in establishing long exact sequences in algebraic topology and their implications for fiber sequences.
  • Mapping cones are significant in establishing long exact sequences because they allow mathematicians to systematically relate different homology groups associated with spaces connected by maps. In fiber sequences, this becomes especially important as it helps depict how fibers vary as we transition through different stages of mapping. By evaluating these relationships, we gain deep insights into how topological spaces interact with each other, particularly regarding their shape and structure across continuous transformations.

Related terms

Homotopy:

A continuous deformation of one function into another, showing that two functions are topologically equivalent.

Fiber sequence: A sequence of topological spaces and continuous maps that generalizes the notion of a fibration, allowing for a way to understand relationships between different spaces.

Suspension:

An operation on a topological space that produces a new space by taking the product of the space with the interval [0, 1] and collapsing the ends.