Tensor Product

Definition

The tensor product is a construction in algebra that combines two vector spaces to create a new vector space, which encapsulates the bilinear interactions between them. This operation allows for a rich interplay of algebraic structures and is foundational in various areas of mathematics, including topology and cohomology theory. It plays a critical role in defining operations such as the cup product and establishing relationships between different spaces, such as when using the Künneth formula.

5 Must Know Facts For Your Next Test

1. The tensor product of two vector spaces $V$ and $W$, denoted $V \otimes W$, results in a new vector space that represents all possible bilinear combinations of elements from $V$ and $W$.
2. In cohomology theory, the cup product is defined using the tensor product, enabling the interaction between different cohomology classes.
3. The Künneth formula utilizes the tensor product to relate the cohomology groups of a product space to the cohomology groups of its factors, allowing for computations in complex spaces.
4. The tensor product is associative, meaning that $(V \otimes W) \otimes U$ is naturally isomorphic to $V \otimes (W \otimes U)$ for any vector spaces $V$, $W$, and $U$.
5. The universal property of the tensor product states that every bilinear map from two vector spaces can be uniquely factored through their tensor product.

Review Questions

• How does the concept of tensor product relate to bilinear maps in vector spaces?
  • The tensor product directly arises from the concept of bilinear maps, as it captures all bilinear interactions between two vector spaces. A bilinear map takes pairs of vectors from each space and produces a scalar in a way that respects linearity. The tensor product provides a way to form a new vector space that embodies these interactions, allowing for further algebraic manipulations and constructions in both algebra and topology.
• In what ways does the cup product utilize tensor products to enhance our understanding of cohomology?
  • The cup product utilizes the tensor product by defining it as an operation on cohomology classes, thereby allowing us to compute new classes from existing ones. Specifically, if we have two cohomology classes represented by cochains, their cup product can be interpreted as an element in the tensor product of their respective cohomology groups. This relationship enriches our understanding by illustrating how different cohomological features interact within a space.
• Evaluate how the Künneth formula uses tensor products to link the cohomological properties of product spaces.
  • The Künneth formula showcases how the tensor product connects the cohomology groups of a product space with those of its constituent spaces. By employing the tensor product, this formula establishes explicit relationships between the individual cohomologies, allowing for easier computation of complex topological structures. The formula emphasizes the role of the tensor product as a bridge between distinct algebraic systems, providing insights into how properties from simpler spaces aggregate into more complex configurations.

Related terms

Bilinear Map: A bilinear map is a function that is linear in each of its arguments separately, allowing it to create a linear combination of inputs from two different vector spaces.

Cohomology: Cohomology is a mathematical tool used to study the properties of topological spaces through algebraic structures, often involving the computation of cohomology groups.

Direct Sum: The direct sum is an operation that combines multiple vector spaces into a new space, where each element is an ordered tuple of elements from the original spaces, highlighting their independent contributions.