## Manifolds modelled on $R^{\infty }$ or bounded weak-* topologies

HTML articles powered by AMS MathViewer

- by Richard E. Heisey
- Trans. Amer. Math. Soc.
**206**(1975), 295-312 - DOI: https://doi.org/10.1090/S0002-9947-1975-0397768-X
- PDF | Request permission

## Abstract:

Let $R^\infty = \lim _ \to R^n$, and let ${B^ \ast }({b^ \ast })$ denote the conjugate, ${B^ \ast }$, of a separable, infinite-dimensional Banach space with its bounded weak-$\ast$ topology. We investigate properties of paracompact, topological manifolds $M,N$ modelled on $F$, where $F$ is either ${R^\infty }$ or ${B^ \ast }({b^ \ast })$. Included among our results are that locally trivial bundles and microbundles over $M$ with fiber $F$ are trivial; there is an open embedding $M \to M \times F$; and if $M$ and $N$ have the same homotopy type, then $M \times F$ and $N \times F$ are homeomorphic. Also, if $U$ is an open subset of ${B^ \ast }({b^ \ast })$, then $U \times {B^ \ast }({b^ \ast })$ is homeomorphic to $U$. Thus, two open subsets of ${B^ \ast }({b^ \ast })$ are homeomorphic if and only if they have the same homotopy type. Our theorems about ${B^ \ast }({b^ \ast })$-manifolds, ${B^ \ast }({b^ \ast })$ as above, immediately yield analogous theorems about $B(b)$-manifolds, where $B(b)$ is a separable, reflexive, infinite-dimensional Banach space with its bounded weak topology.## References

- R. D. Anderson and T. A. Chapman,
*Extending homeomorphisms to Hilbert cube manifolds*, Pacific J. Math.**38**(1971), 281–293. MR**319204** - R. D. Anderson and R. Schori,
*Factors of infinite-dimensional manifolds*, Trans. Amer. Math. Soc.**142**(1969), 315–330. MR**246327**, DOI 10.1090/S0002-9947-1969-0246327-5 - T. A. Chapman,
*Locally-trivial bundles and microbundles with infinite-dimensional fibers*, Proc. Amer. Math. Soc.**37**(1973), 595–602. MR**312524**, DOI 10.1090/S0002-9939-1973-0312524-X - Tammo tom Dieck,
*Partitions of unity in homotopy theory*, Compositio Math.**23**(1971), 159–167. MR**293625** - James Dugundji,
*Topology*, Allyn and Bacon, Inc., Boston, Mass., 1966. MR**0193606** - Nelson Dunford and Jacob T. Schwartz,
*Linear Operators. I. General Theory*, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR**0117523** - Vagn Lundsgaard Hansen,
*Some theorems on direct limits of expanding sequences of manifolds*, Math. Scand.**29**(1971), 5–36. MR**319206**, DOI 10.7146/math.scand.a-11031 - Richard E. Heisey,
*Contracting spaces of maps on the countable direct limit of a space*, Trans. Amer. Math. Soc.**193**(1974), 389–411. MR**367908**, DOI 10.1090/S0002-9947-1974-0367908-6 - Richard E. Heisey,
*Partitions of unity and a closed embedding theorem for $(C^{p},b^*)$-manifolds*, Trans. Amer. Math. Soc.**206**(1975), 281–294. MR**397767**, DOI 10.1090/S0002-9947-1975-0397767-8 - David W. Henderson,
*Micro-bundles with infinite-dimensional fibers are trivial*, Invent. Math.**11**(1970), 293–303. MR**282380**, DOI 10.1007/BF01403183 - David W. Henderson,
*Stable classification of infinite-dimensional manifolds by homotopy-type*, Invent. Math.**12**(1971), 48–56. MR**290413**, DOI 10.1007/BF01389826 - Sze-tsen Hu,
*Theory of retracts*, Wayne State University Press, Detroit, 1965. MR**0181977** - J. L. Kelley and Isaac Namioka,
*Linear topological spaces*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W. R. Scott, Kennan T. Smith. MR**0166578** - Richard S. Palais,
*Banach manifolds of fiber bundle sections*, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 243–249. MR**0448405**

## Bibliographic Information

- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**206**(1975), 295-312 - MSC: Primary 58B05; Secondary 58C20
- DOI: https://doi.org/10.1090/S0002-9947-1975-0397768-X
- MathSciNet review: 0397768