Hopf constructions and higher projective planes for iterated loop spaces
Authors:
Nicholas J. Kuhn, Michael Slack and Frank Williams
Journal:
Trans. Amer. Math. Soc. 347 (1995), 12011238
MSC:
Primary 55P35; Secondary 55P45, 55P47, 55S12
MathSciNet review:
1282890
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Abstract 
References 
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Abstract: We define a category, (for each and ), of spaces with strong homotopy commutativity properties. These spaces have just enough structure to define the DyerLashof operations for fold loop spaces. The category is very convenient for applications since its objects and morphisms are defined in a homotopy invariant way. We then define a functor, , from to the homotopy category of spaces and show to be left adjoint to the fold loop space functor. We then show how one can exploit this adjointness in cohomological calculations to yield new results about iterated loop spaces.
 [Ad]
J. F. Adams, On the nonexistence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20104.
 [AK]
S. Araki and T. Kudo, Topology of spaces and squaring operations, Mem. Fac. Sci. Kyushu Univ. Ser. A 10 (1956), 85120.
 [AS]
M. Arkowitz and P. Silberbush, Some properties of Hopftype constructions, preprint.
 [BV]
J. M. Boardman and R. Vogt, Homotopy everything spaces, Bull. Amer. Math. Soc. 74 (1968), 11171122.
 [Br]
W. Browder, Homology operations and loop spaces, Illinois J. Math. 4 (1960), 347357.
 [BT]
W. Browder and E. Thomas, On the projective plane of an space, Illinois J. Math. 7 (1963), 492502.
 [CLM]
F. R. Cohen, T. J. Lada, and J. P. May, The homology of iterated loop spaces, Lecture Notes in Math., vol. 533, Springer, Berlin and New York, 1976.
 [CMT]
F. R. Cohen, J. P. May, and L. R. Taylor, Splitting of certain spaces , Math. Proc. Cambridge Philos. Soc. 84 (1978), 465496.
 [DL]
E. Dyer and R. Lashof, Homology of iterated loop spaces, Amer. J. Math. 84 (1962), 3588.
 [He]
Y. Hemmi, The projective plane of an pairing, J. Pure Appl. Algebra 75 (1991), 277296.
 [Hu]
J. R. Hubbuck, On homotopy commutative spaces, Topology 8 (1969), 119126.
 [Ka]
R. Kane, Implications in Morava theory, Mem. Amer. Math. Soc. 340 (1986).
 [Ku1]
N. Kuhn, The geometry of the JamesHopf maps, Pacific J. Math. 102 (1982), 397412.
 [Ku2]
, Extended powers of spectra and a generalized KahnPriddy theorem, Topology 23 (1985), 473480.
 [Li]
J. P. Lin, Two torsion and the loop space conjecture, Ann. of Math. (2) 115 (1982), 3591.
 [Mh]
M. Mahowald, The metastable homotopy of , Mem. Amer. Math. Soc. 72 (1967).
 [Ma]
J. P. May, The geometry of iterated loop spaces, Lecture Notes in Math., vol. 271, Springer, Berlin and New York, 1972.
 [Mg1]
R. J. Milgram, Iterated loop spaces, Ann. of Math. (2) 84 (1966), 386403.
 [Mg2]
, Unstable homotopy from the stable point of view, Lecture Notes in Math., vol. 368, Springer, Berlin and New York, 1974.
 [Mi]
H. R. Miller, A spectral sequence for the homology of an infinite delooping, Pacific J. Math. 79 (1978), 139155.
 [Ni]
G. Nishida, Cohomology operations in iterated loop spaces, Proc. Japan Acad. 44 (1968), 104109.
 [S11]
M. Slack, Maps between iterated loop spaces, J. Pure Appl. Algebra 73 (1991), 181201.
 [S12]
, Infinite loop spaces with trivial DyerLashof operations, Math. Proc. Cambridge Philos. Soc. (1993).
 [Sf]
J. Stasheff, spaces from the homotopy point of view, Lectures Notes in Math., vol. 161, Springer, Berlin and New York, 1970.
 [St]
N. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967), 133152.
 [Th1]
E. Thomas, On functional cup operations and the transgression operator, Arch. Math. (Basel) 12 (1961), 435444.
 [Th2]
, Steenrod squares and spaces. I, II, Ann. of Math. (2) 77 (1963), 306317; 81 (1965), 473495.
 [Ad]
 J. F. Adams, On the nonexistence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20104.
 [AK]
 S. Araki and T. Kudo, Topology of spaces and squaring operations, Mem. Fac. Sci. Kyushu Univ. Ser. A 10 (1956), 85120.
 [AS]
 M. Arkowitz and P. Silberbush, Some properties of Hopftype constructions, preprint.
 [BV]
 J. M. Boardman and R. Vogt, Homotopy everything spaces, Bull. Amer. Math. Soc. 74 (1968), 11171122.
 [Br]
 W. Browder, Homology operations and loop spaces, Illinois J. Math. 4 (1960), 347357.
 [BT]
 W. Browder and E. Thomas, On the projective plane of an space, Illinois J. Math. 7 (1963), 492502.
 [CLM]
 F. R. Cohen, T. J. Lada, and J. P. May, The homology of iterated loop spaces, Lecture Notes in Math., vol. 533, Springer, Berlin and New York, 1976.
 [CMT]
 F. R. Cohen, J. P. May, and L. R. Taylor, Splitting of certain spaces , Math. Proc. Cambridge Philos. Soc. 84 (1978), 465496.
 [DL]
 E. Dyer and R. Lashof, Homology of iterated loop spaces, Amer. J. Math. 84 (1962), 3588.
 [He]
 Y. Hemmi, The projective plane of an pairing, J. Pure Appl. Algebra 75 (1991), 277296.
 [Hu]
 J. R. Hubbuck, On homotopy commutative spaces, Topology 8 (1969), 119126.
 [Ka]
 R. Kane, Implications in Morava theory, Mem. Amer. Math. Soc. 340 (1986).
 [Ku1]
 N. Kuhn, The geometry of the JamesHopf maps, Pacific J. Math. 102 (1982), 397412.
 [Ku2]
 , Extended powers of spectra and a generalized KahnPriddy theorem, Topology 23 (1985), 473480.
 [Li]
 J. P. Lin, Two torsion and the loop space conjecture, Ann. of Math. (2) 115 (1982), 3591.
 [Mh]
 M. Mahowald, The metastable homotopy of , Mem. Amer. Math. Soc. 72 (1967).
 [Ma]
 J. P. May, The geometry of iterated loop spaces, Lecture Notes in Math., vol. 271, Springer, Berlin and New York, 1972.
 [Mg1]
 R. J. Milgram, Iterated loop spaces, Ann. of Math. (2) 84 (1966), 386403.
 [Mg2]
 , Unstable homotopy from the stable point of view, Lecture Notes in Math., vol. 368, Springer, Berlin and New York, 1974.
 [Mi]
 H. R. Miller, A spectral sequence for the homology of an infinite delooping, Pacific J. Math. 79 (1978), 139155.
 [Ni]
 G. Nishida, Cohomology operations in iterated loop spaces, Proc. Japan Acad. 44 (1968), 104109.
 [S11]
 M. Slack, Maps between iterated loop spaces, J. Pure Appl. Algebra 73 (1991), 181201.
 [S12]
 , Infinite loop spaces with trivial DyerLashof operations, Math. Proc. Cambridge Philos. Soc. (1993).
 [Sf]
 J. Stasheff, spaces from the homotopy point of view, Lectures Notes in Math., vol. 161, Springer, Berlin and New York, 1970.
 [St]
 N. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967), 133152.
 [Th1]
 E. Thomas, On functional cup operations and the transgression operator, Arch. Math. (Basel) 12 (1961), 435444.
 [Th2]
 , Steenrod squares and spaces. I, II, Ann. of Math. (2) 77 (1963), 306317; 81 (1965), 473495.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199512828909
PII:
S 00029947(1995)12828909
Article copyright:
© Copyright 1995
American Mathematical Society
