Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-13T00:44:07.188Z Has data issue: false hasContentIssue false
  • English
  • Français

Comparing homotopy categories

Published online by Cambridge University Press:  30 November 2007

David Blanc
David Blanc
Affiliation:
[email protected] of Mathematics, University of Haifa, 31905 Haifa, Israel
Get access

Abstract

Given a suitable functor T : → between model categories, we define a long exact sequence relating the homotopy groups of any X ε with those of TX, and use this to describe an obstruction theory for lifting an object G ε to . Examples include finding spaces with given homology or homotopy groups.

Keywords

model categoryhomotopical algebraobstruction theoryrealization
Type
Research Article
Information
Journal of K-Theory , Volume 2 , Issue 1 , August 2008 , pp. 169 - 205
Copyright
Copyright © ISOPP 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

AHK.Aoki, K., Honma, E., & Kaneko, T., “On natural systems of some spaces”, Proc. Jap. Acad. 32 (1956), pp. 564–567Google Scholar
AC.Arkowitz, M. & Curjel, C.R., “The Hurewicz homomorphism and finite homotopy invariants”, Trans. AMS 110 (1964), pp. 538–551CrossRefGoogle Scholar
Ba1.Baues, H.-J., “Relationen für primäre Homotopieoperationen und eine verallge-meinerte EHP-Sequenz”, Ann. Sc. Éc. Norm. Sup. 8 (1975), pp. 509–533CrossRefGoogle Scholar
Ba2.Baues, H.-J., Combinatorial Homotopy and 4-Dimensional Complexes, Gruyter Expositions in Mathematics 2, Walter de Gruyter, Berlin-New York, 1991Google Scholar
Ba3.Baues, H.-J., Homotopy type and homology, Oxford University Press, New York, 1996CrossRefGoogle Scholar
Ba4.Baues, H.-J., Combinatorial Foundation of Homology and Homotopy, Applications to Spaces, Diagrams, Transformation Groups, Compactifications, Differential Algebras, Algebraic Theories, Simplicial Objects, and Resolutions. Springer Monographs in Mathematics, Berlin-New York, 1999Google Scholar
Be.Benkhalifa, M., “Sur le type d'homotopie d'un CW-complexe”, Homology, Homotopy Appl. 5 (2003), pp. 101–120CrossRefGoogle Scholar
BH.Berstein, I. & Hilton, P.J., “On suspensions and comultiplications”, Topology 2 (1963), pp. 73–82CrossRefGoogle Scholar
Bl1.Blanc, D., “New model categories from old”, J. Pure & Appl. Alg. 109 (1996), pp. 37–60CrossRefGoogle Scholar
Bl2.Blanc, D., “CW simplicial resolutions of spaces, with an application to loop spaces”, Top. & Appl. 100 (2000), pp. 151–175CrossRefGoogle Scholar
Bl3.Blanc, D., “Generalized André-Quillen cohomology”, preprint, 2007Google Scholar
BDG.Blanc, D., Dywer, W.G., & Goerss, P.G., “The realization space of a Π-algebra: a moduli problem in algebraic topology”, Topology 43 (2004), pp. 857–892CrossRefGoogle Scholar
BJT.Blanc, D., Johnson, M.J., & Turner, J.M., “On realizing diagrams of Π-algebras”, Algebraic & Geometric Topology 6 (2006), pp. 763–807CrossRefGoogle Scholar
BP.Blanc, D. & Peschke, G., “The fiber of functors between categories of algebras”, J. Pure Appl. Alg. 207 (2006), pp. 687–715CrossRefGoogle Scholar
Bor.Borceux, F., Handbook of Categorical Algebra, Vol. 2: Categories and Structures, Encyc. Math. & its Appl. 51, Cambridge U. Press, Cambridge, UK, 1994Google Scholar
Bou.Bousfield, A.K., “Cosimplicial resolutions and homotopy spectral sequences in model categories”, Geom. & Topology 7 (2003), pp. 1001–1053CrossRefGoogle Scholar
BK.Bousfield, A.K. & Kan, D.M., Homotopy Limits, Completions, and Localizations, Springer Lec. Notes Math. 304, Berlin-New York, 1972Google Scholar
D.Dold, A., “Homology of symmetric products and other functors of complexes”, Ann. Math., Ser. 2 , 68 (1958), pp. 54–80CrossRefGoogle Scholar
DK1.Dwyer, W.G. & Kan, D.M., “A classification theorem for diagrams of simplicial sets”, Topology 23 (1984), pp. 139–155CrossRefGoogle Scholar
DK2.Dwyer, W.G. & Kan, D.M., “An obstruction theory for diagrams of simplicial sets”, Proc. Kon. Ned. Akad. Wet. – Ind. Math. 46 (1984), pp. 139–146Google Scholar
DKS1.Dwyer, W.G., Kan, D.M. & Stover, C.R., “An E 2 model category structure for pointed simplicial spaces”, J. Pure & Appl. Alg. 90 (1993), pp. 137–152CrossRefGoogle Scholar
DKS2.Dwyer, W.G., Kan, D.M. & Stover, C.R. “The bigraded homotopy groups πi,j X of a pointed simplicial space”, J. Pure Appl. Alg. 103 (1995), pp. 167–188CrossRefGoogle Scholar
G.Ganea, T., “A generalization of the homology and homotopy suspensions”, Comm. Math. Helv. 39 (1965), pp. 295–322CrossRefGoogle Scholar
GH.Goerss, P.G. & Hopkins, M.J., “Moduli spaces of commutative ring spectra”, in Baker, A. and Richter, B., eds., Structured Ring Spectra, London Math. Soc. Lect. Notes 315, Cambridge U. Press, Cambridge, 2004, pp. 151–200Google Scholar
Hi.Hirschhorn, P.S., Model Categories and their Localizations, Math. Surveys & Monographs 99, AMS, Providence, RI, 2002Google Scholar
Ho.Hopf, H., “über die Topologie der Gruppen-Mannigfaltkeiten und ihre Verallge-meinerungen”, Ann. Math. (2) 42 (1941), pp. 22–52CrossRefGoogle Scholar
J.Jardine, J.F., “Bousfield's E 2 Model Theory for Simplicial Objects”, in Goerss, P.G. & Priddy, S.B., eds., Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-Theory, Contemp. Math. 346, AMS, Providence, RI 2004, pp. 305–319Google Scholar
Mc.Mac Lane, S., Homology, Springer-Verlag Grund. math. Wissens. 114, Berlin-New York 1963Google Scholar
Man.Mandell, M.A., “E ∞-algebras and p-adic homotopy theory”, Topology 40 (2001), pp.. 43–94CrossRefGoogle Scholar
May.May, J.P., Simplicial Objects in Algebraic Topology, U. Chicago Press, Chicago-London, 1967Google Scholar
Ne.Neisendorfer, J.A., Primary homotopy theory, AMS Memoirs 232, Am. Math. Soc., Providence, RI, 1980Google Scholar
No.Nomura, Y., “On extensions of triads”, Nagoya Math. J. 27 (1966), pp. 249–277CrossRefGoogle Scholar
Q1.Quillen, D.G., Homotopical Algebra, Springer Lec. Notes Math. 43, Berlin-New York, 1963Google Scholar
Q2.Quillen, D.G., “Rational homotopy theory”, Ann. Math. 90 (1969), pp. 205–295CrossRefGoogle Scholar
Q3.Quillen, D.G., “On the (co-)homology of commutative rings”, Applications of Categorical Algebra, Proc. Symp. Pure Math. 17, AMS, Providence, RI, 1970, pp. 65–87Google Scholar
R.Robinson, C.A., “Obstruction theory and the strict associativity of Morava K-theories”, in Salamon, S.M., Steer, B., & Sutherland, W.A., eds., Advances in Homotopy Theory (Cortona, 1988), London Math. Soc. Lect. Notes 139, Cambridge U. Press, Cambridge, 1989, pp. 143–152Google Scholar
Se.Serre, J.-P., “Groupes d'homotopie et classes de groupes abéliens”, Ann. Math. (2) 58 (1953), pp. 258–294CrossRefGoogle Scholar
Sm.Smith, J.R., “Topological realizations of chain complexes I: the general theory”, Top. & Appl. 22 (1986), pp. 301–313CrossRefGoogle Scholar
Sp.Spaliński, J., “Stratified model categories”, Fund. Math. 178 (2003), pp. 217–236CrossRefGoogle Scholar
Sta.Stasheff, J.D., “Homotopy associativity of H-spaces, I,II”, Trans. AMS 108 (1963) pp. 275–292, 293–312Google Scholar
Ste.Steenrod, N.E., “The cohomology algebra of a space”, Ens. Math. 7 (1961), pp. 153–178Google Scholar
Sto.Stover, C.R., “A Van Kampen spectral sequence for higher homotopy groups”, Topology 29 (1990), pp. 9–26CrossRefGoogle Scholar
Sug.Sugawara, M., “A condition that a space is group-like”, Math. J. Okayama U. 7 (1957), pp. 123–149Google Scholar
Sul.Sullivan, D.P., Geometric Topology, Part I. Localization, periodicity, and Galois symmetry, Massachusetts Institute of Technology, Cambridge, MA, 1970Google Scholar
T.Thom, R., “Sur un problème de Steenrod”, C.R. Acad. Set, Paris 236 (1953), pp. 1128–1130Google Scholar
W1.Whitehead, J.H.C., “On simply connected, 4-dimensional polyhedra”, Comm. Math. Helv. 22 (1949), pp. 48–92CrossRefGoogle Scholar
W2.Whitehead, J.H.C., “On the realizability of homotopy groups”, Ann. Math. (2) 50 (1949), pp. 261–263CrossRefGoogle Scholar
W3.Whitehead, J.H.C., “The secondary boundary operator”, Proc. Nat. Acad. Sci. USA 36 (1950), pp. 55–60CrossRefGoogle ScholarPubMed
W4.Whitehead, J.H.C., “A certain exact sequence”, Ann. Math. (2) 52 (1950), pp. 51–110CrossRefGoogle Scholar
W5.Whitehead, J.H.C., “Simple homotopy types”, Amer. J. Math. 72 (1952), pp. 1–57CrossRefGoogle Scholar