Knoerrer Periodicity and Bott Periodicity

TL;DR

This paper explores the connection between Knoerrer periodicity in algebra and Bott periodicity in topology, establishing an 8-periodic version for real hypersurface singularities and linking algebraic invariants to topological K-theory.

Contribution

It introduces an 8-periodic version of Knoerrer periodicity for real hypersurface singularities and constructs a homomorphism connecting algebraic and topological K-theory.

Findings

Proves 8-periodic Knoerrer periodicity for real hypersurface singularities

Constructs a homomorphism from matrix factorizations to topological K-theory

Establishes compatibility between algebraic and topological periodicities

Abstract

The goal of this article is to explain a precise sense in which Knoerrer periodicity in commutative algebra and Bott periodicity in topological K-theory are compatible phenomena. Along the way, we prove an 8-periodic version of Knoerrer periodicity for real isolated hypersurface singularities, and we construct a homomorphism from the Grothendieck group of the homotopy category of matrix factorizations of a complex (real) polynomial $f$ into the topological K-theory of its Milnor fiber (positive or negative Milnor fiber).