Abstract
On the category of smooth manifolds we consider bivariant theories in the sense of Fulton and MacPherson, and give a geometric description of the bivariant topological theory. We show that the bivariant cobordism has a universal property. We also sketch a future application to the space of smooth maps between manifolds.
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References
Atiyah, M. F. and Hirzebruch, F.: Riemann-Roch theorems for differentiable manifolds, Bull. Amer. Math. Soc. 65 (1959), 276–281.
Baum, P. F. and Block, J.: Equivariant bicycles on singular spaces, C. R. Acad. Sci. Paris Ser. I 311(2) (1990), 115–120.
Baum, P. F. and Block, J.: Excess intersection in equivariant bivariant KK-theory, C. R. Acad. Sci. Paris Ser. I 314(5) (1992), 387–392.
Connes, A.: Noncommutative Geometry, Academic Press, 1994.
Dold, A.: Geometric Cobordism and the Fixed Point Transfer, Lecture Notes in Mathematics, Vol. 673, Springer, 1976, pp. 32–87.
Dold, A.: Relations between ordinary and extraordinary cohomology, Coll. Algebraic Topology, August 1-10, 1962, Aarhus University, 1962, pp. 2–9.
Dyer, E.: Cohomology Theories, Benjamin, 1969.
Fulton, W. and MacPherson, R.: Categorical framework for the study of singular spaces, Memoirs of the AMS 243 (1981).
Hirsch, M.: Differential Topology, Springer, Reprint, 1994.
Hirzebruch, F., Berger, T. and Jung, R.: Manifolds and Modular Forms, 2nd edn, Aspects of Mathematics, Vol. E 20, Viehweg, 1994.
Jakob, M.: A Bordism type description of homology, Manuscr. Math. 5 (1998), 61–75.
May, J. P.: E∞Ring Spaces and E∞Ring Spectra, Lecture Notes in Mathematics, Vol. 577, Springer, 1977.
Rudyak, Y. B.: On Thom Spectra, Orientability, and Cobordism, Springer Monographs in Mathematics, Springer, 1998.
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Jakob, M. Bivariant Theories for Smooth Manifolds. Applied Categorical Structures 10, 279–290 (2002). https://doi.org/10.1023/A:1015225622516
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DOI: https://doi.org/10.1023/A:1015225622516