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Construction of Transverse Fields

Published online by Cambridge University Press:  20 November 2018

H. Putz
H. Putz*
Affiliation:
Temple University, Philadelphia, Pennsylvania
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In this paper we give local conditions for a rectilinear embedding of a non-bounded combinatorial manifold, Mn, in Euclidean space, which are sufficient to prove that Mn has a transverse field (see 1.1 and 1.2, definitions).

In a sequel to this paper (6), we will show how with this transverse field we can construct a normal microbundle for the embedded manifold Mn.

Our object in this research was only to obtain an existence theorem for normal microbundles. However, the method of proof via the construction of a transverse field yields as corollaries by Cairns (1), Whitehead (9), or Tao (8), results on smoothing. Earlier smoothing results achieved by the construction of transverse fields in the special case of (global) codimension 1 were obtained by Noguchi (5), and Tao (7; 8).

After the research for this paper was completed, a paper of Davis (2) came to our attention.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1146 - 1159
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Cairns, S. S., Homeomorphisms between topological manifolds and analytic manifolds, Ann. of Math. (2) 41 (1940), 796–808.Google Scholar
2. Davis, H., Manifolds with local codimension one, Thesis, University of Illinois, 1965 (to appear).Google Scholar
3. Hurewicz, W. and Wallman, H., Dimension theory (Princeton Mathematical Series, Vol. 4, Princeton Univ. Press, Princeton, 1941).Google Scholar
4. Munkres, J. R., Elementary differential topology (Princeton Univ. Press, Princeton, N.J., 1963).Google Scholar
5. Noguchi, H., The smoothing of combinatorial n-manifolds in (n + 1) space, Ann. of Math. (2) 72 (1960), 201–215.Google Scholar
6. Putz, H., Transverse field implies normal microbundle, Proc. Amer. Math. Soc. 23 (1969), 232–236.Google Scholar
7. Tao, J., Some properties of (n — \)-manifolds in the Euclidean n-space, Osaka Math. J. 10 (1958), 137–146.Google Scholar
8. Tao, J., On the smoothing of a combinatorial n-manifold immersed in the Euclidean (n + 1)- space, Osaka Math. J. 13 (1961), 229–249.Google Scholar
9. Whitehead, J. H. C., Manifolds with transverse fields in Euclidean space, Ann. of Math. (2) 73 (1961), 154–211.Google Scholar