Detecting mapping spaces

Bittner, Alyson

doi:10.1090/proc/14905

Detecting mapping spaces
HTML articles powered by AMS MathViewer

by Alyson Bittner PDF

Proc. Amer. Math. Soc. 148 (2020), 2683-2693 Request permission

Abstract:

We show if $A$ is a finite CW-complex such that algebraic theories detect mapping spaces out of $A$, then $A$ has the homology type of a wedge of spheres of the same dimension. Furthermore, if $A$ is simply connected, then $A$ has the homotopy type of a wedge of spheres.

References

Bernard Badzioch, Algebraic theories in homotopy theory, Ann. of Math. (2) 155 (2002), no. 3, 895–913. MR 1923968, DOI 10.2307/3062135
Bernard Badzioch, From $\Gamma$-spaces to algebraic theories, Trans. Amer. Math. Soc. 357 (2005), no. 5, 1779–1799. MR 2115076, DOI 10.1090/S0002-9947-04-03711-0
Bernard Badzioch, Kuerak Chung, and Alexander A. Voronov, The canonical delooping machine, J. Pure Appl. Algebra 208 (2007), no. 2, 531–540. MR 2277692, DOI 10.1016/j.jpaa.2006.01.014
Bernard Badzioch and Wojciech Dorabiała, A note on localizations of mapping spaces, Israel J. Math. 177 (2010), 441–444. MR 2684429, DOI 10.1007/s11856-010-0054-5
J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, Berlin-New York, 1973. MR 0420609
A. K. Bousfield, The simplicial homotopy theory of iterated loop spaces, Manuscript, 1992.
A. K. Bousfield, Localization and periodicity in unstable homotopy theory, J. Amer. Math. Soc. 7 (1994), no. 4, 831–873. MR 1257059, DOI 10.1090/S0894-0347-1994-1257059-7
Carles Casacuberta, José L Rodrıguez, and Jin-Yen Tai, Localizations of abelian Eilenberg–Mac Lane spaces of finite type, preprint (1998).
Emmanuel Dror Farjoun, Cellular spaces, null spaces and homotopy localization, Lecture Notes in Mathematics, vol. 1622, Springer-Verlag, Berlin, 1996. MR 1392221, DOI 10.1007/BFb0094429
Philip S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99, American Mathematical Society, Providence, RI, 2003. MR 1944041, DOI 10.1090/surv/099
J. P. May, The geometry of iterated loop spaces, Lecture Notes in Mathematics, Vol. 271, Springer-Verlag, Berlin-New York, 1972. MR 0420610
Haynes Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. (2) 120 (1984), no. 1, 39–87. MR 750716, DOI 10.2307/2007071
Matthew Sartwell, Detecting Mapping Spaces and Derived Equivalence of Algebraic Theories, ProQuest LLC, Ann Arbor, MI, 2016. Thesis (Ph.D.)–State University of New York at Buffalo. MR 3553567
Graeme Segal, Categories and cohomology theories, Topology 13 (1974), 293–312. MR 353298, DOI 10.1016/0040-9383(74)90022-6
James Dillon Stasheff, Homotopy associativity of $H$-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 275-292; ibid. 108 (1963), 293–312. MR 0158400, DOI 10.1090/S0002-9947-1963-0158400-5