Attaching Maps

Definition

Attaching maps are specific continuous functions used in Morse theory to connect manifolds or spaces by gluing them together along common boundaries. They play a crucial role in the construction of the Morse-Smale complex, allowing for the combination of sublevel sets and their critical points to create a coherent structure that reflects the topology of the underlying manifold.

5 Must Know Facts For Your Next Test

1. Attaching maps help define how sublevel sets can be glued together based on the critical points of a Morse function.
2. These maps are crucial in constructing cell complexes, as they determine the way in which new cells are added to an existing structure.
3. The use of attaching maps simplifies the study of homology and cohomology by providing a clear method to build new spaces from simpler pieces.
4. In a Morse-Smale complex, attaching maps are used to connect stable and unstable manifolds associated with critical points.
5. The proper choice of attaching maps ensures that the resulting space maintains desired topological properties, such as being Hausdorff and locally compact.

Review Questions

• How do attaching maps contribute to the construction of a Morse-Smale complex?
  • Attaching maps play an essential role in building a Morse-Smale complex by defining how sublevel sets can be glued together at their critical points. These maps provide a systematic way to combine different regions of the manifold while preserving their topological features. By connecting stable and unstable manifolds associated with critical points, attaching maps ensure that the entire structure accurately represents the underlying topology.
• Discuss the relationship between attaching maps and critical points in Morse theory.
  • In Morse theory, critical points are where a Morse function's derivative vanishes and are significant for understanding the topology of a manifold. Attaching maps directly relate to these critical points by guiding how sublevel sets can be combined based on these specific locations. When constructing complexes, attaching maps help to outline how different regions associated with critical points are integrated into a larger framework, enabling insights into both local and global topological properties.
• Evaluate the impact of choosing appropriate attaching maps on the topological properties of constructed spaces within Morse-Smale complexes.
  • Choosing suitable attaching maps is crucial for preserving the desired topological properties when constructing spaces within Morse-Smale complexes. The right choice ensures that glued sublevel sets maintain important characteristics such as being Hausdorff and locally compact. If attaching maps are improperly defined, it can lead to inconsistencies or undesired properties in the final space, which may hinder further analysis in homology and cohomology. Thus, careful consideration in selecting these maps is vital for ensuring accurate representations of the manifold's topology.

Related terms

Morse Functions: Smooth functions from a manifold to the real numbers that have non-degenerate critical points, which are essential for analyzing the topology of the manifold.

Critical Points: Points in the domain of a Morse function where the derivative vanishes, providing key information about the topology and shape of the manifold.

Sublevel Sets: The sets of points where a Morse function takes values less than or equal to a given threshold, used to study the topology of manifolds.