The homotopy type of Fredholm manifolds

Mukherjea, Kalyan K.

doi:10.1090/S0002-9947-1970-0259954-4

The homotopy type of Fredholm manifolds
HTML articles powered by AMS MathViewer

by Kalyan K. Mukherjea PDF

Trans. Amer. Math. Soc. 149 (1970), 653-663 Request permission

Abstract:

Banach manifolds whose tangent bundles admit a reduction to the Fredholm group have been intensively studied in the last few years. Here we show that such a manifold (under appropriate smoothness and separability restrictions) is homotopy equivalent to the union of a nested sequence of closed finite-dimensional submanifolds.

References

Ralph Abraham and Joel Robbin, Transversal mappings and flows, W. A. Benjamin, Inc., New York-Amsterdam, 1967. An appendix by Al Kelley. MR 0240836
Dan Burghelea and Nicolaas H. Kuiper, Hilbert manifolds, Ann. of Math. (2) 90 (1969), 379–417. MR 253374, DOI 10.2307/1970743
James Eells Jr., Fredholm structures, Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 1, Chicago, Ill., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 62–85. MR 0293676
J. Eells and K. D. Elworthy, Open embeddings of certain Banach manifolds, Ann. of Math. (2) 91 (1970), 465–485. MR 263120, DOI 10.2307/1970634
K. D. Elworthy, Fredholm maps and $\textrm {GL}_{c}(E)$-structures, Bull. Amer. Math. Soc. 74 (1968), 582–586. MR 224113, DOI 10.1090/S0002-9904-1968-12018-X

Infinite-dimensional K-theory

Serge Lang, Introduction to differentiable manifolds, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155257
J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331

Fredholm structures and cohomology

Kalyan K. Mukherjea, Cohomology theory for Banach manifolds, J. Math. Mech. 19 (1969/1970), 731–744. MR 0258069
Frank Quinn, Transversal approximation on Banach manifolds, Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 213–222. MR 0264713
Joel W. Robbin, On the existence theorem for differential equations, Proc. Amer. Math. Soc. 19 (1968), 1005–1006. MR 227583, DOI 10.1090/S0002-9939-1968-0227583-0
S. Smale, An infinite dimensional version of Sard’s theorem, Amer. J. Math. 87 (1965), 861–866. MR 185604, DOI 10.2307/2373250

A Brouwer degree for completely orientable Banach manifolds