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ISSN 1088-6826(online) ISSN 0002-9939(print)



A closed model category
for ($n-1$)-connected spaces

Authors: J. Ignacio Extremiana Aldana, L. Javier Hernández Paricio and M. Teresa Rivas Rodríguez
Journal: Proc. Amer. Math. Soc. 124 (1996), 3545-3553
MSC (1991): Primary 55P15, 55U35
MathSciNet review: 1353370
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Abstract: For each integer $n > 0$, we give a distinct closed model category structure to the category of pointed spaces $ \mathop {\mathrm {Top}}\nolimits _{\star }$ such that the corresponding localized category $ \mathop {\mathrm {Ho}}\nolimits ( \mathop {\mathrm {Top}}\nolimits _{\star }^{n})$ is equivalent to the standard homotopy category of $(n-1)$-connected CW-complexes. The structure of closed model category given by Quillen to $ \mathop {\mathrm {Top}}\nolimits _{\star }$ is based on maps which induce isomorphisms on all homotopy group functors $\pi _{q}$ and for any choice of base point. For each $n>0$, the closed model category structure given here takes as weak equivalences those maps that for the given base point induce isomorphisms on $\pi _{q}$ for $q\ge n$ .

References [Enhancements On Off] (What's this?)

  • [1] J.G. Cabello, A.R. Garzón, Closed model structures for algebraic models of $n$-types, J. Pure Appl. Algebra 103 (1995), 287--302.
  • [2] C. Elvira, L.J. Hernández, Closed model categories for the $n$-type of spaces and simplicial sets, Math. Proc. Camb. Phil. Soc 118 (1995), 93-103. CMP 95:12
  • [3] P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer-Verlag New York, Inc., New York, 1967. MR 0210125
  • [4] A.R. Garzón, J.G. Miranda, Homotopy types of simplicial groups and related closed model structures, Preprint (1994).
  • [5] L. J. Hernández and T. Porter, Categorical models of 𝑛-types for pro-crossed complexes and ℐ_{𝓃}-prospaces, Algebraic topology (San Feliu de Guíxols, 1990) Lecture Notes in Math., vol. 1509, Springer, Berlin, 1992, pp. 146–185. MR 1185969, 10.1007/BFb0087509
  • [6] Timothy Porter, 𝑛-types of simplicial groups and crossed 𝑛-cubes, Topology 32 (1993), no. 1, 5–24. MR 1204402, 10.1016/0040-9383(93)90033-R
  • [7] Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432
  • [8] Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MR 0258031

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Additional Information

J. Ignacio Extremiana Aldana
Affiliation: Departamento de Matemáticas y Computación, Universidad de La Rioja, 26004 Logroño, Spain

L. Javier Hernández Paricio
Affiliation: Departamento de Matemáticas, Universidad de Zaragoza, 50009 Zaragoza, Spain

M. Teresa Rivas Rodríguez
Affiliation: Departamento de Matemáticas y Computación, Universidad de La Rioja, 26004 Logroño, Spain

Keywords: Closed model category, homotopy category, $(n-1)$-connected space
Received by editor(s): May 5, 1995
Additional Notes: The authors acknowledge the financial aid given by the U.R., I.E.R. and DGICYT, project PB93-0581-C02-01.
Communicated by: Thomas Goodwillie
Article copyright: © Copyright 1996 American Mathematical Society