Join

Definition

In topology, the join of two topological spaces is a way to combine them into a new space. Specifically, the join of two spaces $X$ and $Y$, denoted $X * Y$, is formed by taking the product space $X \times Y$ and then collapsing all points of the form $(x,y)$ where $x \in X$ and $y \in Y$ to a single point, typically denoted as the apex. This construction allows for a smooth way to incorporate both spaces into a single structure, playing a significant role in the study of CW complexes and Morse functions.

5 Must Know Facts For Your Next Test

1. The join operation produces a new space that geometrically represents paths connecting points in both original spaces.
2. The join of two spaces can be visualized as taking each point in one space and connecting it to every point in the other space, forming a sort of 'cone-like' structure.
3. Join is associative, meaning that for any three spaces $X$, $Y$, and $Z$, we have $(X * Y) * Z \cong X * (Y * Z)$.
4. The join of a space with a point gives a cone over that space, illustrating how this operation builds upon existing structures.
5. In the context of Morse functions, joins help in constructing CW complexes by allowing for smooth transitions between different dimensional cells.

Review Questions

• How does the join operation combine two topological spaces, and what geometric intuition can be derived from this construction?
  • The join operation combines two topological spaces by taking all points from both and connecting them in a way that encapsulates paths between the two. Geometrically, it can be imagined as forming a cone-like shape where each point from one space connects to every point in the other. This results in a new topological space that retains properties from both original spaces while allowing for complex interactions between them.
• Discuss the role of join in forming CW complexes from Morse functions and how this impacts our understanding of topology.
  • Join plays a crucial role in forming CW complexes by providing a method to incorporate different dimensional cells through smooth transitions. When using Morse functions, which have well-defined critical points, we can create CW complexes that accurately reflect the topology of the underlying manifold. This combination allows us to better understand how spaces behave under various transformations, enabling deeper insights into their topological properties.
• Evaluate how the properties of the join operation can lead to new insights in algebraic topology and its applications.
  • The properties of the join operation, such as associativity and its ability to create cones, can lead to significant insights in algebraic topology by illustrating how different spaces interact and build upon each other. For example, understanding joins helps in calculating homology groups and establishing relationships between various topological constructs. These insights have practical applications in fields such as data analysis and robotics, where understanding complex shapes and their properties is essential.

Related terms

CW Complex: A CW complex is a type of topological space that is built from cells of various dimensions, allowing for a systematic way to analyze spaces through their building blocks.

Morse Function: A Morse function is a smooth function from a manifold to the real numbers that has non-degenerate critical points, which help in understanding the topology of the manifold.

Product Space: The product space is a fundamental construction in topology formed from two topological spaces, where the points are ordered pairs consisting of one point from each space.