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The self-intersection sets of immersed      n n      -manifolds in      ( n + 1 ) (n + 1)      -space provide invariants of the      n n      th stable stem and the      ( n + 1 ) (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 ) (n + 1)      -tuple and      n n      -tuple points, provided the      Z / 2 {\mathbf {Z}}/2      -valued      ( n + 1 ) (n + 1)      -tuple point invariant vanishes      ( n ≥  5 ) (n \geq 5)      . This invariant agrees with the Kervaire invariant for      n = 4 k + 1 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.

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