Join

Definition

In the context of simplicial complexes, a join is a construction that combines two simplicial complexes into a new one by taking the Cartesian product of their vertex sets and forming new simplices from pairs of simplices from each complex. This operation not only retains the structure of the original complexes but also enriches the new complex by adding dimensions, allowing for the exploration of higher-dimensional properties and relationships.

5 Must Know Facts For Your Next Test

1. The join of two simplicial complexes A and B, denoted as A * B, results in a new complex where each simplex in A is connected to each simplex in B, forming higher-dimensional simplices.
2. The dimension of the join of two simplicial complexes is the sum of their dimensions plus one: if A has dimension m and B has dimension n, then A * B has dimension m + n + 1.
3. Joining a simplex with itself creates a complex that includes all possible combinations of its vertices, effectively enriching its structure.
4. The join operation is associative, meaning that the order in which you join multiple complexes does not affect the final result.
5. The join operation helps in visualizing more complex relationships between topological spaces and can be used to study properties like homotopy and homology.

Review Questions

• How does the join operation affect the dimensionality of simplicial complexes?
  • The join operation increases the dimensionality of simplicial complexes by combining them in such a way that every simplex from one complex interacts with every simplex from another. Specifically, if one complex has dimension m and another has dimension n, their join will have dimension m + n + 1. This dimensional increase allows for richer structures and relationships within the new combined complex.
• Discuss the significance of the associative property of the join operation when combining multiple simplicial complexes.
  • The associative property of the join operation means that when combining several simplicial complexes, it does not matter how they are grouped during the joining process. This is significant because it simplifies calculations and theoretical explorations within cohomology and topology, allowing mathematicians to focus on the properties and relationships formed by the overall structure rather than the specific order of operations.
• Evaluate how the concept of join contributes to understanding higher-dimensional topology in relation to simplicial complexes.
  • The concept of join is crucial for understanding higher-dimensional topology because it allows for the construction of new complexes that explore relationships between lower-dimensional spaces. By connecting simplices in a way that enhances their dimensionality, mathematicians can investigate properties like homotopy and homology more effectively. The ability to create more complex structures through join operations ultimately aids in revealing deeper insights into topological spaces and their interconnectedness.

Related terms

Simplicial Complex: A collection of simplices that satisfies certain conditions, such as closure under taking faces and inclusion; this forms the building blocks of topological spaces.

Vertex: A fundamental unit in a simplicial complex, representing a point in space; vertices combine to form higher-dimensional simplices like edges and triangles.

Suspension: An operation on a simplicial complex that creates a new complex by taking the join of a complex with a one-point set, effectively adding an additional dimension.