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Algebraic cycles and infinite loop spaces

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In this paper we use recent results about the topology of Chow varieties to answer an open question in infinite loop space theory. That is, we construct an infinite loop space structure on a certain product of Eilenberg-MacLane spaces so that the total Chern map is an infinite loop map. An analogous result for the total Stiefel-Whitney map is also proved. Further results on the structure of stabilized spaces of alebraic cycles are obtained and computational consequences are also outlined.

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References

  • [AJ] Atiyah, M.F., Jones, J.D.S.: Topological aspects of Yang-Mills theory. Commun. Math. Phys.61, 97–118 (1978)

    Google Scholar 

  • [BHMM] Boyer, C.P., Hurtbise, J.C., Mann, B.M., Milgram, R.J.: The topology of instanton moduli spaces. I. The Atiyah-Jones conjecture. Ann. Math. (to appear)

  • [BM 1] Boyer, C.P., Mann, B.M.: Homology operations on instantons. J. Differ. Geom.28, 423–465 (1988)

    Google Scholar 

  • [BM 2] Boyer, C.P., Mann, B.M.: Monopoles, non-linear σ-models, and two-fold loop spaces. Commun. Math. Phys.115, 571–594 (1988)

    Google Scholar 

  • [BV] Boardman, J.M., Vogt, R.M.: Homotopy invariant algebraic structures on topological spaces. (Lect. Notes Math. Berlin Heidelberg vol. 347) New York: Springer 1973

    Google Scholar 

  • [CCMM] Cohen, F.R., Cohen, R.L., Mann, B.M., Milgram, R.J.: The topology of rational functions and divisors of surfaces. Acta Math.166 (no. 3), 163–221 (1991)

    Google Scholar 

  • [CLM] Cohen, F.R., Lada, T.J., May, J.P.: The homology of, iterated loop spaces. (Lect Notes Math., vol. 533) Berlin Heidelberg New York: Springer 1976

    Google Scholar 

  • [DL] Dyer, E., Lashof, R.K., Homology of iterated loop spaces. Am. J. Math.84, 35–88 (1962)

    Google Scholar 

  • [Don] Donaldson, S.K.: Nahm's equations and the classification of monopoles. Commun. Math. Phys.96, 387–407 (1984)

    Google Scholar 

  • [Gro] Grothendieck, A.: La théorie des classes de Chern. Bull. Soc. Math. Fr.86, 137–154 (1958)

    Google Scholar 

  • [KL] Kraines, D., Lada, T.: A counter-example to the transfer conjecture. In Hoffman, P., Snaith V. (eds.) Algebraic Topology, Waterloo (New York). vol. 741, (Lect. Notes Math., pp. 588–624) Berlin Heidelberg New York: Springer 1979

    Google Scholar 

  • [Koc] Kochman, S.: Homology of the classical groups over the Dyer-Lashof algebra. Trans. Am. Math. Soc.185, 83–136 (1973)

    Google Scholar 

  • [Koz 1] Kozlowski, A.: The Evans-Kahn formula for the total Stiefel-Whitney class. Proc. Am. Math. Soc.91, 309–313 (1984)

    Google Scholar 

  • [Koz 2] Kozlowski, A.: Transfer in the group of multiplicative units of the classical cohomology rings and Stiefel-Whitney classes. Publ. Res. Inst. Math. Sci.25, 59–74 (1989)

    Google Scholar 

  • [Lam] Lam, T.K.: Spaces of real algebraic cycles and homotopy theory Ph.D. thesis, SUNY at Stony Brook, New York (1990)

    Google Scholar 

  • [Law] Lawson, H.B., Jr.: Algebraic cycles and homotopy theory. Ann. Math. II. Ser.129, 253–291 (1989)

    Google Scholar 

  • [LF] Lima-Filho, P.C.: Completions and fibrations for topological monoids. Trans. Am. Math. Soc. (to appear)

  • [LM1] Lawson, H.B., Jr., Michelsohn, M.-L.: Algebraic cycles, Bott periodicity, and the Chern characteristic map. In: Wells, R.O. Jr. (ed.) The mathematical heritage of H. Weyl. (Proc. Symp. Pure Math., vol. 48, pp. 241–264) Providence, RI: Am. Math. Soc. 1988

    Google Scholar 

  • [LM2] Lawson, H.B. Jr., Michelsohn, M.L.: Algebraic cycles, and group actions. In: Differential Geometry, Harlow: Longman 1991, pp. 261–278.

    Google Scholar 

  • [May 1] May, J.P.: The geometry of iterated loop spaces. (Lect. Notes Math., vol. 271 Berlin Heidelberg New York: Springer 1972

    Google Scholar 

  • [May 2] May, J.P.:E -ring spaces andE -ring spectra. (Lect. Notes Math., vol. 577) Berlin Heidelberg New York: Springer 1977

    Google Scholar 

  • [MM] Mann, B.M., Milgram, R.J.: Some spaces of holomorphic maps to complex Grassmann manifolds. J. Differ. Geom33, 301–324 (1991)

    Google Scholar 

  • [Seg 1] Segal, G. The multiplicative group of classical cohomology. Q. J. Math. Oxf. II. Ser.26, 289–293 (1975)

    Google Scholar 

  • [Seg 1] Segal, G.: The topology of rational functions. Acta Math.143, 39–72 (1979)

    Google Scholar 

  • [Sna] Snaith, V.P.: The total Chern and Stiefel-Whitney classes are not infinite loop maps. Ill, J. Math.21, 300–304 (1977)

    Google Scholar 

  • [Ste 1] Steiner, R.: Decompositions of groups of units in ordinary cohomology. Q. J. Math., Oxf.90 (no. 120), 483–494 (1979)

    Google Scholar 

  • [Ste 2] Steiner, R.: Infinite loop structures on the algebraick-theory of spaces. Math. Proc. Camb. Philos. Soc.1 (no. 90), 85–111 (1981)

    Google Scholar 

  • [Tau] Taubes, C.H.: The stable topology of self-dual moduli spaces. J. Differ. Geom.29, 163–230 (1989)

    Google Scholar 

  • [Tot 1] Totaro, B.: The total Chern class is not a map of multiplicative cohomology theories. Math. Z. (to appear)

  • [Tot 2] Totaro, B.: The map from the Chow variety of cycles of degree 2 to the space of all cycles. Math. Sci. Res. Inst. (Preprint, 1990)

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Oblatum XII-1991 & 4-II-1993

All authors were partially supported by the NSF

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Boyer, C.P., Lawson, H.B., Lima-Filho, P. et al. Algebraic cycles and infinite loop spaces. Invent Math 113, 373–388 (1993). https://doi.org/10.1007/BF01244311

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