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

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