Profinite tensor powers

TL;DR

This paper introduces a new way to define tensor products of profinite vector spaces under specific conditions, with applications to Heegaard Floer homology computations.

Contribution

It proposes a novel tensor product construction for profinite vector spaces with finite group actions, applicable to Heegaard Floer homology.

Findings

Defines a tensor product $igotimes_X^{ ext{mcc}} V$ for finite-dimensional $V$ over $ extbf{F}_2$ with profinite index set $X$.

Organizes computations in Heegaard Floer homology related to pro-3-manifolds.

Provides variants for bimodules over products of $ extbf{F}_2$.

Abstract

We discuss the problem of defining a tensor product of profinitely many copies of a vector space $V$, and propose a definition $\bigotimes_X^{\mathrm{mcc}} V$ in the special situation that (1) $V$ is finite-dimensional over $\mathbf{F}_2$, and (2) the profinite $X$ indexing the tensor factors is acted on with finitely many orbits by a pro-$2$-group. The "mcc" on the tensor sign stands for "magnetized and conditionally convergent." A variant construction makes sense when $V$ is a bimodule over a ring of the form $\mathbf{F}_2 \times \cdots \times \mathbf{F}_2$, and the index set $X$ has the profinite version of a cyclic order. The definition organizes some computations in Heegaard Floer homology: it can be pitched as a computation of the Heegaard Floer theory of some pro-$3$-manifolds, though we do not know how to define such a thing.