A congruence for the signature of an embedded manifold

Little, Robert D.

doi:10.1090/S0002-9939-1991-1049846-4

A congruence for the signature of an embedded manifold
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by Robert D. Little

Proc. Amer. Math. Soc. 112 (1991), 587-596

DOI: https://doi.org/10.1090/S0002-9939-1991-1049846-4

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Abstract:

Let ${M^{2n}}$ be a smooth, closed, orientable $2n$-manifold and suppose that $K_x^{2n - 2}$ is an orientable submanifold of ${M^{2n}}$ dual to a cohomology class $x$. If $d$ is a positive integer, the signatures of $K_{dx}^{2n - 2}$ and $K_x^{2n - 2}$ are related by a numerical congruence. If $n$ is odd, then any codimension 2 submanifold of ${\mathbf {C}}{P^n}$ fixed by a diffeomorphism of odd prime order is dual to the generator of the cohomology algebra.

References

Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
G. Brumfiel, On integral $\textrm {PL}$ characteristic classes, Topology 8 (1969), 39–46. MR 234484, DOI 10.1016/0040-9383(69)90029-9
Karl Heinz Dovermann, Rigid cyclic group actions on cohomology complex projective spaces, Math. Proc. Cambridge Philos. Soc. 101 (1987), no. 3, 487–507. MR 878897, DOI 10.1017/S030500410006686X
Akio Hattori, Genera of ramified coverings, Math. Ann. 195 (1972), 208–226. MR 290399, DOI 10.1007/BF01419595
Dale Husemoller, Fibre bundles, 2nd ed., Graduate Texts in Mathematics, No. 20, Springer-Verlag, New York-Heidelberg, 1975. MR 0370578
Robert D. Little, Homotopy complex projective spaces with divisible splitting invariants, Pacific J. Math. 140 (1989), no. 2, 251–262. MR 1023789

The defect of codimension two submanifolds of homotopy complex projective space

Mikiya Masuda, Smooth group actions on cohomology complex projective spaces with a fixed point component of codimension $2$, A fête of topology, Academic Press, Boston, MA, 1988, pp. 585–602. MR 928419
John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554
Ted Petrie, Smooth $S^{1}$ actions on homotopy complex projective spaces and related topics, Bull. Amer. Math. Soc. 78 (1972), 105–153. MR 296970, DOI 10.1090/S0002-9904-1972-12898-2

Triangulating and smoothing homotopy equivalences and homeomorphisms

René Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86 (French). MR 61823, DOI 10.1007/BF02566923
Emery Thomas and John Wood, On manifolds representing homology classes in codimension $2$, Invent. Math. 25 (1974), 63–89. MR 383438, DOI 10.1007/BF01389998
C. T. C. Wall, Surgery on compact manifolds, London Mathematical Society Monographs, No. 1, Academic Press, London-New York, 1970. MR 0431216