Geodesic
The MOD 2 K-Homology of Ω3 S 3 X
Canadian journal of mathematics, Tome 40 (1988) no. 1, pp. 142-196

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In order to compute the group K *(Ω3 S 3 X; Z/2) when X is a finite, torsion free CW-complex we apply the techniques developed by Snaith in [38], [39], [40], [41] which were used in [42] to determine the Atiyah-Hirzebruch spectral sequence ( [11], [1, Part III]) for X as above. Roughly speaking the method consists in defining certain classes in K *(Ω3 S 3 X; Z/2) via the π-equivariant mod 2 K-homology of S 2 × Y 2, ([35]), π the cyclic group of order 2 (acting antipodally on S 2, by permuting factors in Y 2, and diagonally on S 2 × Y 2), Y a finite subcomplex of Ω3 S 3 X, and then showing that the classes so produced map under the edge homomorphism to cycles (in the E 1-term of the Atiyah-Hirzebruch spectral sequence for which determine certain homology classes of H *(Ω3 S 3 X; Z/2), thus exhibiting these as infinite cycles of the spectral sequence 
Mayorquin, J. G. The MOD 2 K-Homology of Ω3 S 3 X. Canadian journal of mathematics, Tome 40 (1988) no. 1, pp. 142-196. doi: 10.4153/CJM-1988-007-2
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[1] 1. Adams, J. F., Lectures on generalized homology, Mathematics Lecture Notes, University of Chicago (1970). Google Scholar

[2] 2. Adem, J., The relations on Steenrodpowers of cohomology classes, Algebraic Geometry and Topology (Princeton University Press, 1957), 191–238. Google Scholar | DOI

[3] 3. Anderson, D. W., Universal coefficient theorems for K-theory (preprint). Google Scholar

[4] 4. Anderson, D. W. and Hodgkin, L., The K-theory of Eilenberg-Mac Lane complexes, Topology. 7 (1968), 317–29. Google Scholar

[5] 5. Araki, S., Hodgkin's theorem, Ann. Math. 85 (1957), 508–525. Google Scholar

[6] 6. Araki, S. and Kudo, T., Topology of H-spaces and H-squaring operations, Mem. Fac. Kyusyu Univ., Ser. A, 10 (1956), 85–120. Google Scholar

[7] 7. Araki, S. and Toda, H., Multiplicative structures in mod q cohomology theories I, II, Osaka J. Math.. 2 (1965), 71–115 and 3 (1966), 81–120. Google Scholar

[8] 8. Araki, S. and Yosimura, Z., Differential Hopf algebras modelled on K-theory mod p I, Osaka J. Math.. 8 (1971), 151–206. Google Scholar

[9] 9. Atiyah, M. F., Characters and cohomology of finite groups, Pub. Math.. 9 (IHES, Paris). Google Scholar

[10] 10. Atiyah, M. F., K-theory (Benjamin Press, 1968). Google Scholar

[11] 11. Atiyah, M. F. and Hirzebruch, F., Vector bundles and homogeneous spaces, Differential Geometry, Proc. of Symp. in Pure Math.. 3 (Amer. Math. Soc, 1961). Google Scholar

[12] 12. Atiyah, M. F. and Segal, G. B., Equivariant K-theory and completion, J. Diff. Geom.. 3 (1969), 1–18. Google Scholar

[13] 13. Boardman, J. M. and Vogt, R. M., Homotopy-everything H-spaces, Bull. Amer. Math. Soc.. 74 (1968), 1117–1122. Google Scholar

[14] 14. Browder, W., Homology operations and loop spaces, Illinois J. Math.. 4 (1960), 347–357. Google Scholar

[15] 15. Brown, E. H. and Peterson, F. P., On the stable decomposition of Ω2Sr+2 , Trans. Am. Math. Soc.. 243(1978), 287–298. Google Scholar

[16] 16. Cartan, H. and Eilenberg, S., Homological algebra (Princeton University Press, 1956). Google Scholar

[17] 17. Caruso, J., Cohen, F. R., May, J. P. and Taylor, L. R., James maps, Segal maps and the Kahn-Priddy theorem, Trans. AMS. 1, 281. Google Scholar

[18] 18. Cohen, J. M., Stable homotopy, Lecture Notes in Math. 165 (Springer-Verlag). Google Scholar

[19] 19. Cohen, F. R., Lada, T. J. and May, J. P., The homology of iterated loop spaces, Lecture Notes in Math.. 533 (1976). Google Scholar | DOI

[20] 20. Cohen, F. R., Mahowald, M. and Milgram, R. J., The stable decomposition of the double loop space of a sphere, AMS Proc. Symp. Pure. Math.. 32 (1978), part 2 (1978), 225–228. Google Scholar

[21] 21. Cohen, F. R., May, J. P. and Taylor, L. R., Splitting of certain spaces CX, Math. Proc. Camb. Phil. Soc. 84 (1978), 465–496. Google Scholar

[22] 22. Dyer, E. and Lashof, R. K., Homology of iterated loop spaces. Amer. J. Math. 84 (1962), 35–88. Google Scholar

[23] 23. Eilenberg, S. and Moore, J. C., Homology and fibrations I, Com. Mat. Helv.. 40 (1966), 199–236. Google Scholar

[24] 24. Hodgkin, L., A Kunneth formula in equivariant K-theory, Warwick Univ. preprint (1968). Google Scholar

[25] 25. Hodgkin, L., The K-theory of Lie groups, Topology. 6 (1967), 1–36. Google Scholar

[26] 26. Hodgkin, L., Dyer-Lashof operations in K-theory, Proc. Oxford Symposium in Algebraic Topology (1972), London Mathematical Society Lecture Note Series 11. Google Scholar

[27] 27. May, J. P., Categories of spectra and infinite loop spaces, Lecture Notes in Math. 99 (Springer-Verlag). Google Scholar

[28] 28. May, J. P., A general algebraic approach to Steenrod operations, Lecture Notes in Math. 168 (Springer-Verlag). Google Scholar

[29] 29. May, J. P., Homology operations on infinite loop spaces, Proc. Symp. Pure Math.. 22 (Amer. Math. Soc. 1971). Google Scholar

[30] 30. May, J. P., The geometry of iterated loop spaces, Lecture Notes in Math. 271 (Springer-Verlag). Google Scholar

[31] 31. May, J. P., E ring spaces and E ring spectra, Lecture Notes in Math. 577 (Springer-Verlag). Google Scholar

[32] 32. Massey, W. S., Exact couples in algebraic topology I, II, Ann. Math.. 56 and 57. Google Scholar

[33] 33. Milnor, J. W., On axiomatic homology theory, Pacific J. Math.. 12 (1962), 337–341. Google Scholar

[34] 34. Rothenberg, M. and Steenrod, N. E., The cohomology of classifying spaces of H-spaces, Bull. Amer. Math. Soc.. 71 (1961). Google Scholar

[35] 35. Segal, G. B., Equivariant K-theory, Pub. Math.. 34 (IHES Paris, 1968). Google Scholar

[36] 36. Segal, G. B., Classifying spaces and spectral sequences, Publ. Math. Inst, des Hautes Etudes Scient. (Paris) 34 (1968). Google Scholar

[37] 37. Snaith, V. P., A stable decomposition of ESX, J. London Math. Soc.. 7 (1974), 577–583. Google Scholar

[38] 38. Snaith, V. P., On the K-theory of homogeneous spaces and conjugate bundles of Lie groups, Proc. L. M. Soc. (3) 22 (1971), 562–584. Google Scholar

[39] 39. Snaith, V. P., On cyclic maps, Proc. Camb. Phil. Soc. (1972), 449–456. Google Scholar

[40] 40. Snaith, V. P., Massey products in K-theory I, II, Proc. Camb. Phil. Soc.. 68 (1970), 303–320 and 69 (1971), 259–289. Google Scholar

[41] 41. Snaith, V. P., Dyer-Lashof operations in K-theory, Lecture Notes in Math.. 496 (1975). Google Scholar

[42] 42. Snaith, V. P., On K*(Ω2X; Z/2), Quart. J. Math. Oxford (3), 26 (1975), 421–436. Google Scholar

[43] 43. Miller, Haines and Snaith, V. P., On K*(Ω2X; Z/2), Canadian Math. Soc. Conference Proc.. 2, Part 1 (1982). Google Scholar

[44] 44. Steenrod, N. E., The cohomology algebra of a space, L'enseignement Math. II Série Tome. 7 (1961), 153–178. Google Scholar

[45] 45. Steenrod, N. E. and Epstein, D. B. A., Cohomology operations, Annals of Math. Study 50 (Princeton Press, 1962). Google Scholar

[46] 46. Whitehead, G. W., Generalized homology theory, Trans. Am. Math Soc.. 102 (1962), 227–283. Google Scholar

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