Mapping spaces and homology isomorphisms
Author:
Nicholas J. Kuhn; \break with an appendix by Greg Arone; Nicholas J. Kuhn
Journal:
Proc. Amer. Math. Soc. 134 (2006), 12371248
MSC (2000):
Primary 55P35; Secondary 55N20, 55P42
Published electronically:
August 29, 2005
MathSciNet review:
2196061
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let denote the space of pointed continuous maps from a finite cell complex to a space . Let be a generalized homology theory. We use Goodwillie calculus methods to prove that under suitable conditions on and , will send an isomorphism in either variable to a map that is monic in homology. Interesting examples arise by letting be theory, the finite complex be a sphere, and the map in the variable be an exotic unstable Adams map between Moore spaces.
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Arone, A generalization of Snaithtype
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 G. Arone, A generalization of Snaithtype filtration, Trans. A. M. S. 351 (1999), 11231250. MR 1638238 (99i:55011)
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 C.F. Bödigheimer, Stable splitting of mapping spaces, Springer L. N. Math. 1286 (1987), 174187. MR 0922926 (89c:55011)
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 A. K. Bousfield, The localization of spaces with respect to homology, Topology 14 (1975), 133150. MR 0380779 (52:1676)
 [B2]
 A. K. Bousfield, On rings and the theory of infinite loop spaces, Theory 10 (1996), no. 1, 130. MR 1373816 (98a:55006)
 [B3]
 A. K. Bousfield, Homotopical localizations of spaces, Amer. J. Math 119 (1997), 13211354. MR 1481817 (98m:55009)
 [EKMM]
 A.D. Elmendorf, I. Kriz, M.A. Mandell, J.P. May, Rings, modules, and algebras in stable homotopy theory, A. M. S. Math. Surveys and Monographs 47, 1997. MR 1417719 (97h:55006)
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 T. G. Goodwillie, Calculus I: the first derivative of pseudoisotopy, Ktheory 4 (1990), 127. MR 1076523 (92m:57027)
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 T. G. Goodwillie, Calculus II: analytic functors, Ktheory 5 (1992), 295332. MR 1162445 (93i:55015)
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 T. G. Goodwillie, Calculus III: the Taylor series of a homotopy functor, Geom. Topol. 7 (2003), 645711. MR 2026544
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 N. J. Kuhn, Suspension spectra and homology equivalences, Trans. A. M. S. 283 (1984), 303313. MR 0735424 (85g:55014)
 [K2]
 N. J. Kuhn, Localization of AndréQuillenGoodwillie towers, and the periodic homology of infinite loopspaces, Advances in Mathematics, to appear (available online).
 [K3]
 N. J. Kuhn, Goodwillie towers and chromatic homotopy: an overview, International Conference in Algebraic Topology (Kinosaki, 2003), Geometry and Topology Monographs, to appear.
 [LS]
 L. Langsetmo and D. Stanley, Nondurable theory equivalence and Bousfield localization, Ktheory 24 (2001), 397410. MR 1885129 (2002k:55008)
 [Ma]
 J. P. May, The geometry of interated loop spaces, Springer L. N. Math. 271, 1972. MR 0420610 (54:8623b)
 [McD]
 D. Mc Duff, Configuration spaces of positive and negative particles, Topology 14 (1975), 91107. MR 0358766 (50:11225)
 [S]
 J. R. Stallings, The embedding of homotopy types into manifolds, unpublished 1965 paper, available at http://math.berkeley.edu/~stall/.
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Additional Information
Nicholas J. Kuhn
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
Email:
njk4x@virginia.edu
DOI:
http://dx.doi.org/10.1090/S0002993905080627
PII:
S 00029939(05)080627
Received by editor(s):
September 2, 2004
Received by editor(s) in revised form:
November 8, 2004
Published electronically:
August 29, 2005
Additional Notes:
This research was partially supported by a grant from the National Science Foundation
Communicated by:
Paul Goerss
Article copyright:
© Copyright 2005 American Mathematical Society
