Proper intersection

Definition

Proper intersection refers to the scenario where two geometric objects intersect transversely, meaning their intersection has the expected dimension based on their dimensions and position in space. This concept is fundamental in understanding how shapes interact in intersection theory, revealing important relationships between their topological properties and dimensions.

5 Must Know Facts For Your Next Test

1. In a proper intersection, if two submanifolds of dimensions $m$ and $n$ intersect in a proper manner, their intersection will be a submanifold of dimension $m+n-d$, where $d$ is the dimension of the ambient space.
2. The notion of proper intersection allows for the formulation of intersection numbers, which count how many times and in what manner two manifolds intersect.
3. Improper intersections occur when the intersection fails to meet the transversality condition, leading to complications in defining properties like dimension and intersection number.
4. Proper intersections are crucial for the application of various results in algebraic geometry and topology, linking geometric properties to algebraic invariants.
5. The study of proper intersections involves tools from both differential geometry and algebraic topology, highlighting its importance across multiple areas of mathematics.

Review Questions

• How does proper intersection differ from improper intersection, and why is this distinction important in intersection theory?
  • Proper intersection occurs when two geometric objects intersect transversely, leading to predictable properties about their intersection. In contrast, improper intersections do not meet this transversality condition, which can complicate our understanding of dimensions and intersections. This distinction is crucial because it influences the application of various mathematical techniques and results that rely on the predictable nature of proper intersections.
• Discuss how the concept of transversality is linked to the idea of proper intersection and its implications in both geometry and topology.
  • Transversality is directly tied to proper intersection as it ensures that two manifolds meet in a well-defined manner. When manifolds intersect transversely, they satisfy certain conditions that allow for straightforward calculation of their dimensions and properties. This connection between transversality and proper intersection underlines their significance in both geometry and topology, enabling mathematicians to use tools from one area to solve problems in the other.
• Evaluate how the study of proper intersections contributes to our understanding of algebraic invariants in topology.
  • The study of proper intersections enhances our understanding of algebraic invariants by linking geometric properties with algebraic structures. When two manifolds intersect properly, we can define intersection numbers and other invariants that are essential for classifying spaces up to homeomorphism or homotopy equivalence. This relationship not only clarifies geometric interactions but also enables mathematicians to apply algebraic methods to derive deeper insights into the structure and behavior of topological spaces.

Related terms

Transversality: A condition that ensures two or more manifolds intersect in a way that is both expected and well-defined, typically resulting in an intersection of the correct dimension.

Homology: A mathematical concept that studies topological spaces through the use of algebraic structures, focusing on the properties that remain invariant under continuous transformations.

Cohomology: An algebraic tool used to derive information about topological spaces by associating algebraic objects, such as groups or rings, with these spaces.