Surgery on codimension one immersions in 𝑅ⁿ⁺¹: removing 𝑛-tuple points

Carter, J. Scott

doi:10.1090/S0002-9947-1986-0857434-8

Surgery on codimension one immersions in $\textbf {R}^ {n+1}$: removing $n$-tuple points
HTML articles powered by AMS MathViewer

by J. Scott Carter

Trans. Amer. Math. Soc. 298 (1986), 83-101

DOI: https://doi.org/10.1090/S0002-9947-1986-0857434-8

PDF | Request permission

Abstract:

The self-intersection sets of immersed $n$-manifolds in $(n + 1)$-space provide invariants of the $n$th stable stem and the $(n + 1)$st stable homotopy of infinite real projective space. Theorems of Eccles [5] and others [1, 8, 14, 19] relate these invariants to classically defined homotopy theoretic invariants. In this paper a surgery theory of immersions is developed; the given surgeries affect the self-intersection sets in specific ways. Using such operations a given immersion may be surgered to remove $(n + 1)$-tuple and $n$-tuple points, provided the ${\mathbf {Z}}/2$-valued $(n + 1)$-tuple point invariant vanishes $(n \geq 5)$. This invariant agrees with the Kervaire invariant for $n = 4k + 1$. These results first appeared in my dissertation [2]; a summary was presented in [3]. Some results and methods have been improved since these works were written. In particular, the proof of Theorem 14 has been simplified.

References

Thomas F. Banchoff, Triple points and surgery of immersed surfaces, Proc. Amer. Math. Soc. 46 (1974), 407–413. MR 377897, DOI 10.1090/S0002-9939-1974-0377897-1

Surgery on immersions: A geometric approach to stable homotopy

Haynes R. Miller and Stewart B. Priddy (eds.), Proceedings of the Northwestern Homotopy Theory Conference, Contemporary Mathematics, vol. 19, American Mathematical Society, Providence, R.I., 1983. Held at Northwestern University, Evanston, Ill., March 22–26, 1982. MR 711037
H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, 4th ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 14, Springer-Verlag, Berlin-New York, 1980. MR 562913, DOI 10.1007/978-3-662-21943-0
Peter John Eccles, Multiple points of codimension one immersions, Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen, Siegen, 1979), Lecture Notes in Math., vol. 788, Springer, Berlin, 1980, pp. 23–38. MR 585650
Peter John Eccles, Multiple points of codimension one immersions of oriented manifolds, Math. Proc. Cambridge Philos. Soc. 87 (1980), no. 2, 213–220. MR 553578, DOI 10.1017/S030500410005667X
Peter John Eccles, Codimension one immersions and the Kervaire invariant one problem, Math. Proc. Cambridge Philos. Soc. 90 (1981), no. 3, 483–493. MR 628831, DOI 10.1017/S0305004100058941
Michael H. Freedman, Quadruple points of $3$-manifolds in $S^{4}$, Comment. Math. Helv. 53 (1978), no. 3, 385–394. MR 505553, DOI 10.1007/BF02566085
D. Hilbert and S. Cohn-Vossen, Geometry and the imagination, Chelsea Publishing Co., New York, N. Y., 1952. Translated by P. Neményi. MR 0046650
Daniel S. Kahn and Stewart B. Priddy, On the transfer in the homology of symmetric groups, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 1, 91–101. MR 464229, DOI 10.1017/S0305004100054323
Daniel S. Kahn and Stewart B. Priddy, On the transfer in the homology of symmetric groups, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 1, 91–101. MR 464229, DOI 10.1017/S0305004100054323
Ulrich Koschorke, Multiple points of immersions, and the Kahn-Priddy theorem, Math. Z. 169 (1979), no. 3, 223–236. MR 554526, DOI 10.1007/BF01214837

Vector fields and other vector-bundle morphisms—a singularity approach

Ulrich Koschorke and Brian Sanderson, Geometric interpretations of the generalized Hopf invariant, Math. Scand. 41 (1977), no. 2, 199–217. MR 474289, DOI 10.7146/math.scand.a-11714
Ulrich Koschorke and Brian Sanderson, Self-intersections and higher Hopf invariants, Topology 17 (1978), no. 3, 283–290. MR 508891, DOI 10.1016/0040-9383(78)90032-0
J. Lannes, Sur les immersions de Boy, Algebraic topology, Aarhus 1982 (Aarhus, 1982) Lecture Notes in Math., vol. 1051, Springer, Berlin, 1984, pp. 263–270 (French). MR 764583, DOI 10.1007/BFb0075571
Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. MR 331377, DOI 10.1007/BF01390197
Hirosi Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies, No. 49, Princeton University Press, Princeton, N.J., 1962. MR 0143217
Pierre Vogel, Cobordisme d’immersions, Ann. Sci. École Norm. Sup. (4) 7 (1974), 317–357 (1975) (French). MR 372878, DOI 10.24033/asens.1272
Robert Wells, Cobordism groups of immersions, Topology 5 (1966), 281–294. MR 196760, DOI 10.1016/0040-9383(66)90011-5

On generalizing Boy’s surface: Constructing a generator of the third stable stem

Invariants of regular homotopy and bordism of low dimensional immersions

George W. Whitehead, Recent advances in homotopy theory, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 5, American Mathematical Society, Providence, R.I., 1970. MR 0309097