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Profinite tensor powers

David Treumann, C. -M. Michael Wong

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.

Profinite tensor powers

Abstract

We discuss the problem of defining a tensor product of profinitely many copies of a vector space , and propose a definition in the special situation that (1) is finite-dimensional over , and (2) the profinite indexing the tensor factors is acted on with finitely many orbits by a pro--group. The "mcc" on the tensor sign stands for "magnetized and conditionally convergent." A variant construction makes sense when is a bimodule over a ring of the form , and the index set 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--manifolds, though we do not know how to define such a thing.

Paper Structure

This paper contains 8 sections, 1 theorem, 15 equations.

Table of Contents

  1. Introduction
  2. Infinite tensor products
  3. Profinite tensor products
  4. Germs of groups and germs of actions
  5. Finite tensor powers of bimodules
  6. Profinite tensor powers of bimodules---the "solenoidal sector"
  7. The staircase picture
  8. An example

Key Result

Theorem 1

Let $G \cong \mathbf{Z}_2$ be the $2$-adic completion of $H_1(M)$, let $X$ be an affine copy of $G$, and let $S\colon X \to X$ be the "obvious" solenoidal structure given by $S(x) = x+1$. There is a semisimple algebra $I$ and an $(I,I)$-bimodule $V$ such that with as in eq:intro-F/F-bimod, there are isomorphisms

Theorems & Definitions (1)

  • Theorem