Oriented cobordism of random manifolds

TL;DR

This paper develops a framework for studying random manifolds within a non-commutative probability setting, introducing oriented cobordism groups and extending characteristic number invariants to the non-commutative case.

Contribution

It introduces the oriented cobordism groups of random manifolds in a non-commutative context and computes these groups in low dimensions, extending classical invariants.

Findings

Computed cobordism groups in dimensions 0 and 1

Proved surjectivity of the Thom-Pontryagin homomorphism

Extended characteristic numbers to non-commutative random manifolds

Abstract

We introduce a general framework allowing the systematic study of random manifolds. In order to do so, we will put ourselves in a more general context than usual by allowing the underlying probability space to be non commutative. We introduce in this paper the oriented cobordism groups of random manifolds, which we compute in dimensions 0 and 1, and we prove the surjectivity of the corresponding Thom-Pontryagin homomorphism. Non commutative and usual (commutative) random manifolds are naturally related in this setting since two commutative random manifolds can be cobordant through a non commutative one, even if they are not cobordant as usual random manifolds. The main interest of our approach is that expected characteristic numbers can be generalized to the non commutative case and remain naturally invariant up to cobordism. This includes Hirzebruch signature, Pontryagin numbers and, more generally, the expected index of random elliptic differential operators