Article
Full entry |
PDF
(0.5 MB)
Feedback
PDF
(0.5 MB)
Feedback
Keywords:
homotopy colimit; cofibrant replacement; effective homology; equivariant
homotopy colimit; cofibrant replacement; effective homology; equivariant
Summary:
We extend the notion of simplicial set with effective homology presented in [22] to diagrams of simplicial sets. Further, for a given finite diagram of simplicial sets $X \colon \mathcal{I}\rightarrow \mbox{sSet}$ such that each simplicial set $X(i)$ has effective homology, we present an algorithm computing the homotopy colimit $\mbox{hocolim}\,X$ as a simplicial set with effective homology. We also give an algorithm computing the cofibrant replacement $X^{\mbox{cof}}$ of $X$ as a diagram with effective homology. This is applied to computing of equivariant cohomology operations.
References:
[1] Bousfield, A.K., Kan, D.M.: Homotopy Limits, Completions and Localizations. Lecture Notes in Math., no. 304, Springer, 1972. DOI 10.1007/978-3-540-38117-4_12 | MR 0365573 | Zbl 0259.55004
[2] Bredon, G.: Equivariant cohomology theories. Lecture Notes in Math., no. 34, Springer-Verlag, Berlin-New York, 1967. MR 0214062 | Zbl 0162.27202
[3] Brown, R.: The twisted Eilenberg-Zilber theorem. 1965 Simposio di Topologia (Messina, 1964), 1965, pp. 33–37. MR 0220273
[4] Čadek, M., Krčál, M., Matoušek, J., Sergeraert, F., Vokřínek, L., Wagner, U.: Computing all maps into a sphere. J. ACM 61 (2014), article no. 17. DOI 10.1145/2597629 | MR 3215297 | Zbl 1295.68196
[5] Čadek, M., Krčál, M., Matoušek, J., Vokřínek, L., Wagner, U.: Polynomial time computation of homotopy groups and Postnikov systems in fixed dimensions. SIAM J. Comput. 43 (2014), 1728–1780. DOI 10.1137/120899029 | MR 3268623
[6] Čadek, M., Krčál, M., Vokřínek, L.: Algorithmic solvability of lifting extension problem. arXiv:1307.6444, 2013.
[7] Dugger, D.: A primer on homotopy colimits. http://math.uoregon.edu/~ddugger/hocolim.pdf, 2008.
[8] Dwyer, W., Hirschhorn, P., Kan, D., Smith, J.: Homotopy Limit Functors on Model Categories and Homotopical Categories. Math. Surveys Monogr., vol. 113, American Mathematical Society, Providence, 2004. MR 2102294 | Zbl 1072.18012
[9] Dwyer, W.G., Kan, D.M.: An obstruction theory for diagrams of simplicial sets. Nederl. Akad. Wetensch. Indag. Math. 46 (2) (1984), 139–146. DOI 10.1016/1385-7258(84)90015-5 | MR 0749527 | Zbl 0555.55018
[10] Eilenberg, S., MacLane, S.: On the groups $H(\pi , n)$, I. Ann. of Math. (2) 58 (1953), 55–106. MR 0056295 | Zbl 0050.39304
[11] Eilenberg, S., MacLane, S.: On the groups $H(\pi , n)$, II. Ann. of Math. (2) 60 (1954), 49–139. MR 0065163 | Zbl 0055.41704
[12] Elmendorf, A.D.: Systems of fixed points sets. Trans. Amer. Math. Soc. 277 (1983), 275–284. DOI 10.1090/S0002-9947-1983-0690052-0 | MR 0690052
[13] Filakovský, M., Vokřínek, L.: Are two maps homotopic? An algorithmic viewpoint. arXiv:1312.2337, 2013.
[14] Goerss, P.G., Jardine, J.F.: Simplicial homotopy theory. Birkhauser, Boston-Basel-Berlin, 1999. MR 1711612 | Zbl 0949.55001
[15] Gugenheim, V.K.A.M.: On the chain-complex of a fibration. Illinois J. Math. 16 (1972), 398–414. MR 0301736 | Zbl 0238.55015
[16] Heras, J.: Effective homology for the pushout of simplicial sets. Proceedings XII Encuentros de Algebra Computacional y Aplicaciones (EACA 2010), 2010, pp. 152–156.
[17] Isaacson, S.B.: Exercises on homotopy colimits. http://math.mit.edu/~mbehrens/TAGS/Isaacson_exer.pdf
[18] Krčál, M., Matoušek, J., Sergeraert, F.: Polynomial-time homology for simplicial Eilenberg-MacLane spaces. arXiv:1201.6222, 2012. MR 3124946 | Zbl 1295.68201
[19] May, J.P.: Simplicial Objects in Algebraic Topology. Chicago Lectures in Math., Univ. Chicago Press, 1992, 1992 reprint of 1967 original. MR 1206474 | Zbl 0769.55001
[20] May, J.P., Piacenza, R.J., Cole, M.: Equivariant homotopy and cohomology theory: Dedicated to the memory of Robert J. Piacenza. Providence, R.I.: Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, 1996. MR 1413302
[21] Riehl, E.: Categorical Homotopy Theory. Cambridge University Press, 2014. MR 3221774
[22] Rubio, J., Sergeraert, F.: Constructive Homological Algebra and Applications. Tech. report, Written in 2006 for a MAP Summer School at the University of Genova, 2012, arXiv:1208.3816v2.
[23] Shih, W.: Homologie des espaces fibrés. Publications Mathématiques de l'IHÉS 13 (1962), 5–87. MR 0144348 | Zbl 0105.16903
[24] Stephan, M.: On equivariant homotopy theory for model categories. arXiv:1308.0856, 2013.
Search
Partner of
