Cohomology with local coefficients and knotted manifolds

We show how the classical notions of cohomology with local coefficients, CW-complex, covering space, homeomorphism equivalence, simple homotopy tubular neighbourhood, and spinning can be encoded on a computer used to calculate ambient isotopy invariants continuous embeddings N ↪ M one topological manifold into another. More specifically, we describe an algorithm for computing homology H n ( X , A ) finite connected CW-complex coefficients in Z π 1 -module when is finitely generated over . It used, particular, compute integral ˜ induced homomorphism → map p : associated index subgroup < as well corresponding homomorphism. illustrate open-source implementation by using it that: (i) degree 2 group distinguishes between types complements ⊂ R 4 spun Hopf link Satoh's tube welded (these two having isomorphic fundamental groups homology); (ii) − B 3 granny knot reef knot, where boundary again homology). Our open source allows user experiment further examples knots, knotted surfaces, other spaces. conclude paper explanation also provides approach set [ W ] ϕ based classes maps f CW-complexes fixed = case dim ⁡ i 0 ≤