Cohomology with local coefficients and knotted manifolds

Abstract

We show how the classical notions of cohomology with local coefficients, CW-complex, covering space, homeomorphism equivalence, simple homotopy equivalence, tubular neighbourhood, and spinning can be encoded on a computer and used to calculate ambient isotopy invariants of continuous embeddings $N\hookrightarrow M$ of one topological manifold into another. More specifically, we describe an algorithm for computing the homology $H_n(X,A)$ and cohomology $H^n(X,A)$ of a finite connected CW-complex X with local coefficients in a $\mathbb Z\pi_1X$-module $A$ when $A$ is finitely generated over $\mathbb Z$. It can be used, in particular, to compute the integral cohomology $H^n(\widetilde X_H,\mathbb Z)$ and induced homomorphism $H^n(X,\mathbb Z) \rightarrow H^n(\widetilde X_H,\mathbb Z)$ for the covering map $p\colon \widetilde X_H \rightarrow X$ associated to a finite index subgroup $H < \pi_1X$, as well as the corresponding homology homomorphism. We illustrate an open-source implementation of the algorithm by using it to show that: (i) the degree $2$ homology group $H_2(\widetilde X_H,\mathbb Z)$ distinguishes between the homotopy types of the complements $X\subset \mathbb R^4$ of the spun Hopf link and Satoh's tube map of the welded Hopf link (these two complements having isomorphic fundamental groups and integral homology); (ii) the degree $1$ homology homomorphism $H_1(p^{-1}(B),\mathbb Z) \rightarrow H_1(\widetilde X_H,\mathbb Z)$ distinguishes between the homeomorphism types of the complements $X\subset \mathbb R^3$ of the granny knot and the reef knot, where $B\subset X$ is the knot boundary (these two complements again having isomorphic fundamental groups and integral homology).