# The Fredholm index for operators of tensor product type

## TL;DR

This paper establishes a topological index theorem for bisingular pseudodifferential operators of tensor product type on product manifolds, linking their Fredholm index to a topological invariant.

## Contribution

It introduces a topological index theorem for tensor product type operators and proves the equality of Fredholm and topological indices for external tensor product operators.

## Key findings

- Proved a topological index theorem for bisingular operators.
- Established the equality of Fredholm and topological indices for tensor product operators.
- Constructed a double deformation groupoid and Poincaré duality homomorphism.

## Abstract

We consider bisingular pseudodifferential operators which are pseudodifferential operators of tensor product type. These operators are defined on the product manifold $M_1 \times M_2$, for closed manifolds $M_1$ and $M_2$. We prove a topological index theorem of product type. In addition, we show that the Fredholm index of elliptic bisingular operators equals the topological index, whenever the operator takes the form of an external tensor product of pseudodifferential operators, up to equivalence. To this end we construct a suitable double deformation groupoid and a Poincar\'e duality type homomorphism.