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Normal Invariants of Lens Spaces

Published online by Cambridge University Press:  20 November 2018

Carmen M. Young
Carmen M. Young*
Affiliation:
Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139-4307 USA
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Abstract

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We show that normal and stable normal invariants of polarized homotopy equivalences of lens spaces $M=\,L({{2}^{m}};\,{{r}_{1}},...,\,{{r}_{n}})$ and $N\,=\,L({{2}^{m}};\,{{s}_{1}},...,{{s}_{n}})$ are determined by certain $\ell $-polynomials evaluated on the elementary symmetric functions ${{\sigma }_{i}}\,(r_{1}^{2},...,r_{n}^{2})$ and ${{\sigma }_{i}}(s_{1}^{2},...,s_{n}^{2})$. Each polynomial ${{\ell }_{k}}$ appears as the homogeneous part of degree $k$ in the Hirzebruch multiplicative $L$-sequence. When $n=8$, the elementary symmetric functions alone determine the relevant normal invariants.

Keywords

57R6557S25
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 41 , Issue 3 , 01 September 1998 , pp. 374 - 384
Copyright
Copyright © Canadian Mathematical Society 1998

References

[A] Adams, J. F., Infinite Loop Spaces. Ann. of Math. Stud. 76, Princeton Univ. Press, 1978.Google Scholar
[BG] Becker, J. C. and Gottlieb, D. H., The transfer map and fiber bundles. Topology 14 (1975), 1–12.Google Scholar
[Br] Browder, W., Surgery on Simply-Connected Manifolds. Ergeb. Math. 65, Springer-Verlag, 1972.Google Scholar
[BPW] Browder, W., Petrie, T. and Wall, C. T. C., The classification of free actions of cyclic groups of odd order on homotopy spheres. Bull. Amer.Math. Soc. 77 (1971), 455–459.Google Scholar
[CS] Cappell, S. E. and Shaneson, J. L., Non-linear similarity. Ann. of Math. 113 (1981), 315–355.Google Scholar
[CSSWW] Cappell, S., Shaneson, J., Steinberger, M., Weinberger, S. and West, J., The topological classification of linear representations of Z2r . Bull. Amer.Math. Soc. 22 (1990), 51–57.Google Scholar
[E] Evens, L., The Cohomology of Groups. Oxford University Press, 1991.Google Scholar
[H] Hirzebruch, F., Topological Methods in Algebraic Geometry. Die Grundlehren der Math. 131, Springer-Verlag, 1966.Google Scholar
[KS] Kirby, R. and Siebenman, L. C., Foundational Essays on Topological Manifolds, Smoothings and Triangulations. Ann. of Math. Stud. 88 (1977).Google Scholar
[MM] Madsen, I. and Milgram, R. J., The Classifying Spaces for Surgery and Cobordism of Manifolds. Ann. of Math. Stud. 92, Princeton Univ. Press, 1979.Google Scholar
[MiSt] Milnor, J. and Stasheff, J., Characteristic Classes. Ann. of Math. Stud. 76, Princeton Univ. Press, 1974.Google Scholar
[MoSu] Morgan, J. and Sullivan, D., The transversality characteristic class and linking cycles in surgery theory. Ann. of Math. 99 (1974), 383–463.Google Scholar
[N] Nicas, A. J., Induction theorems for groups of homotopy manifold structures. Mem. Amer.Math. Soc. 267 (1982).Google Scholar
[O] Olum, P., Mappings of manifolds and the notion of degree. Ann. of Math. 58 (1953), 458–480.Google Scholar
[Su] Sullivan, D., Geometric topology, part I. Localisation, periodicity and Galois symmetry. Mimeographed notes, MIT, 1970.Google Scholar
[Sz] Szczarba, R. H., On tangent bundles of fibre spaces and quotient spaces. Amer. J. Math. 86 (1964), 685–697.Google Scholar
[tDH] Dieck, T. Tom and Hambleton, I., Surgery Theory and Geometry of Representations. Birkhauser Verlag, 1988.Google Scholar
[W] Wall, C. T. C., Surgery on Compact Manifolds. Academic Press, 1970.Google Scholar