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- W3038961350 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 $Nhookrightarrow M$ 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 $mathbb Zpi_1X$-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 $pcolon widetilde X_H rightarrow X$ 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 $Xsubset mathbb R^4$ 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 $Xsubset mathbb R^3$ of the granny knot and the reef knot, where $Bsubset X$ is the knot boundary (these two complements again having isomorphic fundamental groups and integral homology)." @default.
- W3038961350 created "2020-07-10" @default.
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- W3038961350 date "2019-11-12" @default.
- W3038961350 modified "2023-10-01" @default.
- W3038961350 title "Cohomology with local coefficients and knotted manifolds" @default.
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