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Transactions of the American Mathematical Society

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



Cobordism invariants, the Kervaire invariant and fixed point free involutions

Author: William Browder
Journal: Trans. Amer. Math. Soc. 178 (1973), 193-225
MSC: Primary 57D90; Secondary 57D65
MathSciNet review: 0324717
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Abstract: Conditions are found which allow one to define an absolute version of the Kervaire invariant in ${Z_2}$ of a ${\text {Wu - }}(q + 1)$ oriented 2q-manifold. The condition is given in terms of a new invariant called the spectral cobordism invariant. Calculations are then made for the Kervaire invariant of the n-fold disjoint union of a manifold M with itself, which are then applied with $M = {P^{2q}}$, the real protective space. These give examples where the Kervaire invariant is not defined, and other examples where it has value $1 \in {{\mathbf {Z}}_2}$. These results are then applied to construct examples of smooth fixed point free involutions of homotopy spheres of dimension $4k + 1$ with nonzero desuspension obstruction, of which some Brieskorn spheres are examples (results obtained also by Berstein and Giffen). The spectral cobordism invariant is also applied directly to these examples to give another proof of a result of Atiyah-Bott. The question of which values can be realized as the sequence of Kervaire invariants of characteristic submanifolds of a smooth homotopy real projective space is discussed with some examples. Finally a condition is given which yields smooth embeddings of homotopy ${P^m}$’s in ${R^{m + k}}$ (which has been applied by E. Rees).

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Keywords: Spectral cobordism invariant, <IMG WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${Z_2}$">-valued quadratic forms and their radicals, Arf invariant, surgery obstructions, characteristic submanifolds of smooth homotopy projective spaces, Brieskorn varieties, embeddings
Article copyright: © Copyright 1973 American Mathematical Society