Smooth structure and signature of codimension 2 embeddings

Erle, Dieter

doi:10.1090/S0002-9939-1972-0303553-X

Smooth structure and signature of codimension $2$ embeddings
HTML articles powered by AMS MathViewer

by Dieter Erle PDF

Proc. Amer. Math. Soc. 35 (1972), 611-616 Request permission

Abstract:

In an earlier paper (Topology 8 (1969), 99-114), the author defined a signature for codimension 2 embeddings in ${S^{4k + 1}}$ and proved this signature to be an invariant of the topological type of the embedding. For embedded homotopy $(4k - 1)$-spheres, this signature is known to detect the smooth structure. It turns out that it determines the smooth structure also for integer homology $(4k - 1)$-spheres and, by a result of A. Durfee, for closed $(2k - 2)$ connected $(4k - 1)$-manifolds with finite $(2k - 1)$-dimensional homology. As a consequence, in the above cases, the smooth structure is given by the topological type of the embedding. On the other hand, for $k = 3,4,5,7,15$, we exhibit examples of $(4k - 1)$ manifolds embedded in a $(4k + 1)$-sphere for which the smooth structure is not determined by the signature of the embedding.

References

J. F. Adams, On the groups $J(X)$. IV, Topology 5 (1966), 21–71. MR 198470, DOI 10.1016/0040-9383(66)90004-8
P. L. Antonelli, On stable diffeomorphism of exotic spheres in the metastable range, Canadian J. Math. 23 (1971), 579–587. MR 283810, DOI 10.4153/CJM-1971-065-x
Fumiko Bandô and Kiyoshi Katase, A remark on the $\pi \$-imbedding of homotopy spheres, Proc. Japan Acad. 45 (1969), 443–445. MR 253351
Edgar H. Brown Jr. and Franklin P. Peterson, The Kervaire invariant of $(8k+2)$-manifolds, Bull. Amer. Math. Soc. 71 (1965), 190–193. MR 170346, DOI 10.1090/S0002-9904-1965-11278-2
R. De Sapio, Differential structures on a product of spheres, Comment. Math. Helv. 44 (1969), 61–69. MR 243536, DOI 10.1007/BF02564512

Diffeomorphism classification of isolated hypersurface singularities

Dieter Erle, Quadratische Formen als Invarianten von Einbettungen der Kodimension $2$, Topology 8 (1969), 99–114 (German). MR 238300, DOI 10.1016/0040-9383(69)90002-0
Michel A. Kervaire and John W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR 148075, DOI 10.1090/S0273-0979-2015-01504-1
R. C. Kirby and L. C. Siebenmann, On the triangulation of manifolds and the Hauptvermutung, Bull. Amer. Math. Soc. 75 (1969), 742–749. MR 242166, DOI 10.1090/S0002-9904-1969-12271-8
J. Levine, A classification of differentiable knots, Ann. of Math. (2) 82 (1965), 15–50. MR 180981, DOI 10.2307/1970561
W. S. Massey, On the normal bundle of a sphere imbedded in Euclidean space, Proc. Amer. Math. Soc. 10 (1959), 959–964. MR 109351, DOI 10.1090/S0002-9939-1959-0109351-8
John W. Milnor and Michel A. Kervaire, Bernoulli numbers, homotopy groups, and a theorem of Rohlin, Proc. Internat. Congress Math. 1958., Cambridge Univ. Press, New York, 1960, pp. 454–458. MR 0121801
James Munkres, Obstructions to the smoothing of piecewise-differentiable homeomorphisms, Ann. of Math. (2) 72 (1960), 521–554. MR 121804, DOI 10.2307/1970228
Reinhard E. Schultz, Smooth structures on $S^{p}\times S^{q}$, Ann. of Math. (2) 90 (1969), 187–198. MR 250321, DOI 10.2307/1970687
Hirosi Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies, No. 49, Princeton University Press, Princeton, N.J., 1962. MR 0143217
C. T. C. Wall, Classification of $(n-1)$-connected $2n$-manifolds, Ann. of Math. (2) 75 (1962), 163–189. MR 145540, DOI 10.2307/1970425
C. T. C. Wall, Classification problems in differential topology. VI. Classification of $(s-1)$-connected $(2s+1)$-manifolds, Topology 6 (1967), 273–296. MR 216510, DOI 10.1016/0040-9383(67)90020-1