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↪M of one topological manifold into another. More specifically, we describe an algorithm for computing the homology Hn(X,A) and cohomology Hn(X,A) of a finite connected CW-complex X with local coefficients in a Zπ1X-module A when A is finitely generated over Z. It can be used, in particular, to compute the integral cohomology Hn(X˜H,Z) and induced homomorphism Hn(X,Z)→Hn(X˜H,Z) for the covering map p:X˜H→X associated to a finite index subgroup H<π1X, 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 H2(X˜H,Z) distinguishes between the homotopy types of the complements X⊂R4 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 H1(p−1(B),Z)→H1(X˜H,Z) distinguishes between the homeomorphism types of the complements X⊂R3 of the granny knot and the reef knot, where B⊂X is the knot boundary (these two complements again having isomorphic fundamental groups and integral homology). Our open source implementation allows the user to experiment with further examples of knots, knotted surfaces, and other embeddings of spaces. We conclude the paper with an explanation of how the cohomology algorithm also provides an approach to computing the set [W,X]ϕ of based homotopy classes of maps f:W→X of finite CW-complexes over a fixed group homomorphism π1f=ϕ in the case where dimW=n, π1X is finite and πiX=0 for 2≤i≤n−1.
Highlights
Let f, g : N → M be two continuous cellular embeddings of a finite CW-complex N into a finite CW-complex M
A continuous cellular map H : M × [0, 1] → M, x → Ht(x) is said to be an ambient isotopy between f and g if H0 is the identity map on M, each Ht is a homeomorphism from M to itself, and H1 ◦ f = g
For embeddings of spheres Sn → Sn+k interest is on the case k = 2 thanks to a result of Zeeman (1961) which states that any two embeddings are ambient isotopic for k ≥ 3, and a result of Mazur (1959) which implies that there is again just one isotopy class of embeddings for k = 1
Summary
Let f, g : N → M be two continuous cellular embeddings of a finite CW-complex N into a finite CW-complex M. The complements Xi = R4 \ Ki have isomorphic fundamental groups and isomorphic integral homology groups It was shown by Kauffman and Faria Martins (2008) that the spaces X1, X2 are homotopy inequivalent; their technique involves the fundamental crossed module derived from the lower dimensions of the universal cover of a space, and counts the representations of this fundamental crossed module into a given finite crossed module. (This easy observation reduces to complexity bounds for Krushkal’s spanning tree algorthm and for the Smith Normal Form algorithm.) one can compute the set [W, X]φ in polynomial time (ignoring practical constraints such as availability of memory) This is a modest addition to deeper results in Cadek et al (2014a), Krazal et al (2013), Cadek et al (2014b) which establish the existence of a polynomial-time algorithm for computing the set [W, X] of homotopy classes of maps W → X under the hypothesis that W, X are simplicial sets, that πiX = 0 for 0 ≤ i ≤ n − 1, and that dim W ≤ 2n − 2.
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