Profinite tensor powers

Abstract

We discuss the problem of defining a tensor product of profinitely many copies of a vector space $V$, and propose a definition ${⨂}_X^{\mathrm{mcc}}V$ 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.

36 pages, 3 figures