The Smith Fiber Sequence and Invertible Field Theories

TL;DR

This paper provides a comprehensive framework for Smith homomorphisms between bordism groups, unifying various examples, and extends them to a long exact sequence with applications to invertible field theories and quantum physics.

Contribution

It introduces a general, equivalent definition of Smith homomorphisms, extends them to a long exact sequence, and explores their implications for invertible field theories and physics.

Findings

Unified the theory of Smith homomorphisms with multiple definitions.

Extended Smith homomorphisms to a long exact sequence of bordism groups.

Connected the sequence to physical theories via Anderson duals and invertible field theories.

Abstract

Smith homomorphisms are maps between bordism groups that change both the dimension and the tangential structure. We give a completely general account of Smith homomorphisms, unifying the many examples in the literature. We provide three definitions of Smith homomorphisms, including as maps of Thom spectra, and show they are equivalent. Using this, we identify the cofiber of the spectrum-level Smith map and extend the Smith homomorphism to a long exact sequence of bordism groups, which is a powerful computation tool. We discuss several examples of this long exact sequence, relating them to known constructions such as Wood's and Wall's sequences. Furthermore, taking Anderson duals yields a long exact sequence of invertible field theories, which has a rich physical interpretation. We developed the theory in this paper with applications in mind to symmetry breaking in quantum field theory, which we study in a companion paper.