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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Surgery on codimension one immersions in $ {\bf R}\sp {n+1}$: removing $ n$-tuple points


Author: J. Scott Carter
Journal: Trans. Amer. Math. Soc. 298 (1986), 83-101
MSC: Primary 57R42; Secondary 55N22, 55Q10, 57R65
MathSciNet review: 857434
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1986-0857434-8
Article copyright: © Copyright 1986 American Mathematical Society