Involutions on 3-manifolds and self-dual, binary codes (Q943562)

A \textit{self-dual, binary code} \(C\) of length \(k\) is a linear subspace of the \(k\)-dimensional vector space \((\mathbb{Z}/2)^{k}\) over the field \(\mathbb{Z}/2,\) which coincides with its orthogonal complement with respect to the standard ``Euclidean'' form \(\langle x,y\rangle:=\sum_{i}x_{i}y_{i} \in\mathbb{Z}/2,\) for \(x=(x_{1},\dots,x_{k}),\) \(y=(y_{1},\dots,y_{k} )\in(\mathbb{Z}/2)^{k}.\) The weight, \(\omega t(x),\) of a code word \(x\in C\) is defined as the number of non-zero coordinates in \(x=(x_{1},\dots,x_{k}).\) Let \(\tau:M\rightarrow M\) be an involution on a closed \(3\)-manifold \(M\) with finitely many fixed points \(x_{1},\dots,x_{k}.\) If \(k=\dim_{\mathbb{Z}/2} (\bigoplus_{i}H^{i}(M;\mathbb{Z}/2)),\) then we say that the number of fixed points is maximal. Notice that, by Smith theory, it is always \(k\leq \dim_{\mathbb{Z}/2}(\bigoplus_{i}H^{i}(M;\mathbb{Z}/2)).\) The authors prove that every involution \(\tau\) with only isolated fixed points on a compact \(3\)-manifold \(M\) determines a self-dual, binary code \(C(M,\tau).\) The code corresponding to such an involution is described, firstly using equivariant cohomology and secondly using the ordinary homology with \(\mathbb{Z}/2\) coefficients of the complement of a neighborhood of the fixed points in the orbit space. In the following, the authors prove that every self-dual, binary code can be obtained from an involution on an orientable \(3\)-manifold with the maximal number of isolated fixed points. A particular interesting class of self-dual, binary codes are \textit{doubly even codes}, i.e. codes whose code words have weights zero mod \(4.\) The authors define the concept of a Spin-involution and relate doubly even codes to Spin manifolds. More precisely, they prove that Spin involutions give doubly even codes and each doubly even code comes from an orientable \(3\)-manifold with Spin-involution with the maximal number of isolated fixed points.