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.
Similar content being viewed by others
Notes
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.
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(t, a) is independent of t, i.e. \(h(t,a)=f(a)\) for all \(t\in [0,1]\) and \(a\in A\).
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.
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.
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.
The complex is (the canonical triangulation of) the union of all products \(\sigma \times \tau \) of disjoint simplices \(\sigma ,\,\tau \in K\), \(\sigma \cap \tau = \emptyset \).
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.
We recall that a contraction is a map \(\sigma \) of degree 1 satisfying \(\partial \sigma +\sigma \partial ={\mathrm {id}}\).
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 \(\pi \).
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.
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.
Thus, the action needs only be free away from A and the same generalization applies to the dimension.
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.
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 \({P_{n}}\rightarrow B\).
References
Bredon, G.E.: Equivariant Cohomology Theories. Lecture Notes in Mathematics, vol. 34. Springer, Berlin (1967)
Brown, E.H.: Finite computability of Postnikov complexes. Ann. Math. 65, 1–20 (1957)
Čadek, M., Krčál, M., Vokřínek, L.: Algorithmic solvability of the lifting-extension problem (extended version). http://arxiv.org/abs/1307.6444 (2013)
Čadek, M., Krčál, M., Matoušek, J., Sergeraert, F., Vokřínek, L., Wagner, U.: Computing all maps into a sphere. J. ACM 61(17), 1–44 (2014)
Čadek, M., Krčál, M., Matoušek, J., Vokřínek, L., Wagner, U.: Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension. SIAM J. Comput. 43, 1728–1780 (2014)
Čadek, M., Krčál, M., Matoušek, J., Vokřínek, L., Wagner, U.: Extendability of continuous maps is undecidable. Discrete Comput. Geom. 51, 24–66 (2014)
Crabb, M., James, I.: Fibrewise Homotopy Theory. Springer Monographs in Mathematics. Springer, London (1998)
Dwyer, W.G., Spalinski, J.: Homotopy theories and model categories. In: James, I.M. (ed.) Handbook of Algebraic Topology, pp. 73–126. Elsevier, Amsterdam (1995)
Eilenberg, S., MacLane, S.: On the groups \(H(\Pi,n)\), II. Methods of computation. Ann. Math. 60, 49–139 (1954)
Filakovský, M.: Effective chain complexes for twisted products. Arch. Math. (Brno) 48, 313–322 (2012)
Filakovský, M.: Algorithmic construction of the Postnikov tower for diagrams of simplicial sets. Thesis. https://is.muni.cz/th/211334/prif_d/THESIS_Marek_Filakovsky.pdf
Filakovský, M., Vokřínek, L.: Are two given maps homotopic? An algorithmic viewpoint. http://arxiv.org/abs/1312.2337 (2013)
Friedman, G.: An elementary illustrated introduction to simplicial sets. Rocky Mt. J. Math. 42, 353–423 (2012)
Hirschhorn, P.S.: Model Categories and Their Localizations. Mathematical Surveys and Monographs, vol. 99. American Mathematical Society, Providence, RI (2003)
Matoušek, J.: Using the Borsuk–Ulam Theorem. Springer, Berlin (2003)
Matoušek, J., Tancer, M., Wagner, U.: Hardness of embedding simplicial complexes in \({\mathbb{R}}^{d}\). J. Eur. Math. Soc. 13, 259–295 (2011)
May, J.P.: Simplicial Objects in Algebraic Topology. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1992)
McClendon, J.F.: Obstruction theory in fiber spaces. Math. Z. 120, 1–17 (1971)
Robinson, C.A.: Moore-Postnikov systems for non-simple fibrations. Ill. J. Math. 16, 234–242 (1972)
Rubio, J., Sergeraert, F.: Constructive homological algebra and applications. http://arxiv.org/abs/1208.3816 (2012) (Written in 2006 for a MAP Summer School at the University of Genova)
Shih, W.: Homologie des espaces fibres. Publ. Math. l’IHÉS 13, 93–176 (1962)
Stasheff, J.: \(H\)-Spaces From a Homotopy Point of View. Lecture Notes in Mathematics, vol. 161. Springer, Berlin (1970)
Vokřínek, L.: Computing the abelian heap of unpointed stable homotopy classes of maps. Arch. Math. (Brno) 49, 359–368 (2013)
Vokřínek, L.: Constructing homotopy equivalences of chain complexes of free \({\mathbb{Z}}G\)-modules. Contemp. Math. 617, 279–296 (2014)
Vokřínek, L.: Heaps and unpointed stable homotopy theory. Arch. Math. (Brno) 50, 323–332 (2014)
Vokřínek, L.: Decidability of the extension problem for maps into odd-dimensional spheres. Discrete Comput. Geom. 57(1), 1–11 (2017)
Weber, C.: Plongements de polyedres dans le domaine metastable. Comment. Math. Helv. 42, 1–27 (1967)
Author information
Authors and Affiliations
Corresponding author
Additional information
Editor in Charge: Günter M. Ziegler
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).
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-016-9855-6
Keywords
- Homotopy classes
- Equivariant
- Fibrewise
- Lifting-extension problem
- Algorithmic computation
- Embeddability
- Moore–Postnikov tower