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Algorithmic Solvability of the Lifting-Extension Problem

Abstract

Let X and Y be finite simplicial sets (e.g. finite simplicial complexes), both equipped with a free simplicial action of a finite group G. Assuming that Y is d-connected and \(\dim X\le 2d\), for some \(d\ge 1\), we provide an algorithm that computes the set of all equivariant homotopy classes of equivariant continuous maps \(|X|\rightarrow |Y|\); the existence of such a map can be decided even for \(\dim X\le 2d+1\). This yields the first algorithm for deciding topological embeddability of a k-dimensional finite simplicial complex into \(\mathbb R^n\) under the condition \(k\le \frac{2}{3} n-1\). More generally, we present an algorithm that, given a lifting-extension problem satisfying an appropriate stability assumption, computes the set of all homotopy classes of solutions. This result is new even in the non-equivariant situation.

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Notes

  1. An extension of [4] to the case of a simply connected Y whose non-stable homotopy groups, i.e. the groups \({\pi _{n}}(Y)\) for \({n}> 2 {d}\), are finite (e.g. an odd-dimensional sphere) that works for X of arbitrary dimension can be found in [26].

  2. It is possible, for a given homotopy class \(z \in [X, Y]\), to go through all subdivisions \(X'\) and all possible simplicial maps \(X' \rightarrow Y\) and test if they represent z. However, such a procedure does not seem to be very effective.

  3. A homotopy \(h :[0,1]\times X\rightarrow Y\) is fibrewise if \(\psi (h(t,x))=g(x)\) for all \(t\in [0,1]\) and \(x\in X\). It is relative to A if, for \(a\in A\), h(ta) is independent of t, i.e. \(h(t,a)=f(a)\) for all \(t\in [0,1]\) and \(a\in A\).

  4. If \(\psi \) is a Kan fibration between finite simply connected simplicial sets then its fibre is a finite Kan complex and it is easy to see that it then must be discrete. Consequently, \(\psi \) is a covering map between simply connected spaces and thus an isomorphism.

  5. The problem of computing homotopy classes of solutions (under our usual condition on the dimension of X) was considered in [5], but with a different equivalence relation on the set of all extensions: [5] dealt with the (slightly unnatural) coarse classification, where two extensions \(\ell _0\) and \(\ell _1\) are considered equivalent if they are homotopic as maps \(X\rightarrow Y\), whereas here we deal with the fine classification, where the equivalence of \(\ell _0\) and \(\ell _1\) means that they are homotopic relative to A.

  6. The homotopy fibre of \(\psi \) is the fibre of \(\psi '\), where \(\psi \) is factored through \(Y'\) as above. It is unique up to homotopy equivalence, and so the connectivity is well defined.

  7. The complex is (the canonical triangulation of) the union of all products \(\sigma \times \tau \) of disjoint simplices \(\sigma ,\,\tau \in K\), \(\sigma \cap \tau = \emptyset \).

  8. These requirements (with the exception of the differentials) are automatically satisfied when the elements of the chain complex are represented directly as \({\mathbb ZG}\)-linear combinations of the distinguished bases.

  9. We recall that a contraction is a map \(\sigma \) of degree 1 satisfying \(\partial \sigma +\sigma \partial ={\mathrm {id}}\).

  10. Our groups \(H_G^*(X;\pi )\) are the equivariant cohomology groups of X with coefficients in a certain system associated with \(\pi \) (see the remark in [1, Sect. I.9]) or, alternatively, they are the cohomology groups of X / G with local coefficients specified by \(\pi \).

  11. Given two such homotopies, one may form out of them a map , whose extension to \(\Delta ^{2}\times X\), fibrewise over B, gives on \(d_2\Delta ^{2}\times X\) the required homotopy.

  12. This is a solution of a lifting extension problem whose left part is an inclusion in the pair \((\Delta ^{1}, \partial \Delta ^{1}) \times (\Delta ^{1}, 0) \times (X, A)\) with the middle term \(\infty \)-connected, thus also the whole product, and the inclusion is a weak homotopy equivalence.

  13. Thus, the action needs only be free away from A and the same generalization applies to the dimension.

  14. The fibres of \(\psi \) are \({n}\)-connected and isomorphic to those of \(\ell ^*P\rightarrow X\). From the long exact sequence of homotopy groups of this fibration, it follows that \(\ell ^*P\rightarrow X\) is also an \(({n}+1)\)-equivalence and its section then must be an \({n}\)-equivalence.

  15. Start with an inclusion \((\Delta ^{{i}}\times *)\cup (0\times F)\rightarrow \sigma ^*{P_{n}}\) given by the zero section on the first summand and by the inclusion on the second. Extend this to a fibrewise map \(\Delta ^{{i}}\times F\rightarrow \sigma ^*{P_{n}}\) which is a fibrewise homotopy equivalence, hence an isomorphism, by the minimality of \({P_{n}}\rightarrow B\).

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Acknowledgements

We are grateful to Jiří Matoušek and Uli Wagner for many useful discussions, comments and suggestions that improved this paper a great deal. Moreover, this paper could hardly exist without our long-term collaboration, partly summarized in [4,5,6].

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Correspondence to Lukáš Vokřínek.

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The research of M. Č. was supported by the Project CZ.1.07/2.3.00/20.0003 of the Operational Programme Education for Competitiveness of the Ministry of Education, Youth and Sports of the Czech Republic. The research by M. K. was supported by the Center of Excellence—Inst. for Theor. Comput. Sci., Prague (Project P202/12/G061 of GA ČR) and by the Project LL1201 ERCCZ CORES. The research of L. V. was supported by the Center of Excellence—Eduard Čech Institute (Project P201/12/G028 of GA ČR).

Editor in Charge: Günter M. Ziegler

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Čadek, M., Krčál, M. & Vokřínek, L. Algorithmic Solvability of the Lifting-Extension Problem. Discrete Comput Geom 57, 915–965 (2017). https://doi.org/10.1007/s00454-016-9855-6

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Keywords

  • Homotopy classes
  • Equivariant
  • Fibrewise
  • Lifting-extension problem
  • Algorithmic computation
  • Embeddability
  • Moore–Postnikov tower

Mathematics Subject Classification

  • Primary 55Q05
  • Secondary 55S91