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Transactions of the American Mathematical Society

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Free and semi-inert cell attachments

Author: Peter Bubenik
Journal: Trans. Amer. Math. Soc. 357 (2005), 4533-4553
MSC (2000): Primary 55P35; Secondary 16E45
Published electronically: June 21, 2005
MathSciNet review: 2156720
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $Y$ be the space obtained by attaching a finite-type wedge of cells to a simply-connected, finite-type CW-complex.

We introduce the free and semi-inert conditions on the attaching map which broadly generalize the previously-studied inert condition. Under these conditions we determine $H_*(\Omega Y;R)$ as an $R$-module and as an $R$-algebra, respectively. Under a further condition we show that $H_*(\Omega Y;R)$ is generated by Hurewicz images.

As an example we study an infinite family of spaces constructed using only semi-inert cell attachments.

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

  • [AH56] J. F. Adams and P. J. Hilton, On the chain algebra of a loop space, Comment. Math. Helv. 30 (1956), 305-330. MR 0077929 (17:1119b)
  • [Ani82] David J. Anick, A counterexample to a conjecture of Serre, Ann. of Math. (2) 115 (1982), no. 1, 1-33. MR 0644015 (86i:55011a)
  • [Ani89] -, Homotopy exponents for spaces of category two, Algebraic topology (Arcata, CA, 1986), Lecture Notes in Math., vol. 1370, Springer, Berlin, 1989, pp. 24-52. MR 1000365 (90c:55010)
  • [Ani92] -, Single loop space decompositions, Trans. Amer. Math. Soc. 334 (1992), no. 2, 929-940. MR 1145728 (93g:55011)
  • [Avr82] Luchezar L. Avramov, Free Lie subalgebras of the cohomology of local rings, Trans. Amer. Math. Soc. 270 (1982), no. 2, 589-608. MR 0645332 (83g:13010)
  • [Bau81] Hans Joachim Baues, Commutator calculus and groups of homotopy classes, London Mathematical Society Lecture Note Series, vol. 50, Cambridge University Press, Cambridge, 1981. MR 0634675 (83b:55012)
  • [Bub03] Peter Bubenik, Cell attachments and the homology of loop spaces and differential graded algebras, Ph.D. thesis, University of Toronto, 2003.
  • [Bub04] -, Free cell attachments and the rational homotopy lie algebra, arXiv:math.AT/0406405, 2004.
  • [FHT84] Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Sur l'homotopie des espaces de catégorie $2$, Math. Scand. 55 (1984), no. 2, 216-228. MR 0787198 (86k:55006)
  • [FHT01] -, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR 1802847 (2002d:55014)
  • [FT89] Yves Félix and Jean-Claude Thomas, Effet d'un attachement cellulaire dans l'homologie de l'espace des lacets, Ann. Inst. Fourier (Grenoble) 39 (1989), no. 1, 207-224. MR 1011984 (90j:55012)
  • [HL87] Stephen Halperin and Jean-Michel Lemaire, Suites inertes dans les algèbres de Lie graduées (``Autopsie d'un meurtre. II''), Math. Scand. 61 (1987), no. 1, 39-67. MR 0929396 (89e:55022)
  • [HL96] Kathryn Hess and Jean-Michel Lemaire, Nice and lazy cell attachments, J. Pure Appl. Algebra 112 (1996), no. 1, 29-39. MR 1402394 (97e:55006)
  • [Lem78] Jean-Michel Lemaire, ``Autopsie d'un meurtre'' dans l'homologie d'une algèbre de chaînes, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 1, 93-100. MR 0500930 (58:18423)
  • [MM65] John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211-264. MR 0174052 (30:4259)
  • [Sco02] Jonathan A. Scott, Algebraic structure in the loop space homology Bockstein spectral sequence, Trans. Amer. Math. Soc. 354 (2002), no. 8, 3075-3084 (electronic). MR 1897391 (2003c:55008)
  • [Sco03] -, A torsion-free Milnor-Moore theorem, J. London Math. Soc. (2) 67 (2003), no. 3, 805-816. MR 1967707 (2004c:55017)
  • [Whi39] J. H. C. Whitehead, Simplicial spaces, nuclei and $m$-groups, Proc. London Math. Soc. (2) 45 (1939), 243-327.
  • [Whi41] -, On adding relations to homotopy groups, Ann. of Math. (2) 42 (1941), 409-428. MR 0004123 (2:323c)

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Additional Information

Peter Bubenik
Affiliation: Institut de Géométrie, Algèbre et Topologie, Ecole Polytechnique Fédérale de Lausanne, EPFL/SB/IGAT (BCH), 1015 Lausanne, Switzerland

Keywords: Cell attachments, loop space, loop space homology, Adams-Hilton models, differential graded algebras, Lie models
Received by editor(s): December 5, 2003
Published electronically: June 21, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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