## Cobordism invariants, the Kervaire invariant and fixed point free involutions

HTML articles powered by AMS MathViewer

- by William Browder PDF
- Trans. Amer. Math. Soc.
**178**(1973), 193-225 Request permission

## 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 2*q*-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).

## References

- J. F. Adams,
*On the groups $J(X)$. II*, Topology**3**(1965), 137–171. MR**198468**, DOI 10.1016/0040-9383(65)90040-6 - J. F. Adams,
*Vector fields on spheres*, Ann. of Math. (2)**75**(1962), 603–632. MR**139178**, DOI 10.2307/1970213 - M. F. Atiyah,
*Thom complexes*, Proc. London Math. Soc. (3)**11**(1961), 291–310. MR**131880**, DOI 10.1112/plms/s3-11.1.291 - M. F. Atiyah and R. Bott,
*A Lefschetz fixed point formula for elliptic complexes. II. Applications*, Ann. of Math. (2)**88**(1968), 451–491. MR**232406**, DOI 10.2307/1970721 - Israel Berstein,
*Involutions with nonzero Arf invariant*, Bull. Amer. Math. Soc.**74**(1968), 678–682. MR**236932**, DOI 10.1090/S0002-9904-1968-11991-3 - Glen E. Bredon,
*Exotic actions on spheres*, Proc. Conf. on Transformation Groups (New Orleans, La., 1967) Springer, New York, 1968, pp. 47–76. MR**0266239** - William Browder,
*The Kervaire invariant of framed manifolds and its generalization*, Ann. of Math. (2)**90**(1969), 157–186. MR**251736**, DOI 10.2307/1970686 - William Browder,
*Surgery on simply-connected manifolds*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 65, Springer-Verlag, New York-Heidelberg, 1972. MR**0358813**, DOI 10.1007/978-3-642-50020-6 - William Browder,
*Embedding smooth manifolds*, Proc. Internat. Congr. Math. (Moscow, 1966) Izdat. “Mir”, Moscow, 1968, pp. 712–719. MR**0238335** - W. Browder and G. R. Livesay,
*Fixed point free involutions on homotopy spheres*, Bull. Amer. Math. Soc.**73**(1967), 242–245. MR**206965**, DOI 10.1090/S0002-9904-1967-11700-2 - Edgar H. Brown Jr.,
*The Arf invariant of a manifold*, Conf. on Algebraic Topology (Univ. of Illinois at Chicago Circle, Chicago, Ill., 1968) Univ. of Illinois at Chicago Circle, Chicago, Ill., 1969, pp. 9–18. MR**0251737** - Michikazu Fujii,
*$K_{O}$-groups of projective spaces*, Osaka Math. J.**4**(1967), 141–149. MR**219060** - Charles H. Giffen,
*Desuspendability of free involutions on Brieskorn spheres*, Bull. Amer. Math. Soc.**75**(1969), 426–429. MR**240823**, DOI 10.1090/S0002-9904-1969-12204-4 - Charles H. Giffen,
*Smooth homotopy projective spaces*, Bull. Amer. Math. Soc.**75**(1969), 509–513. MR**239607**, DOI 10.1090/S0002-9904-1969-12223-8 - André Haefliger and Valentin Poenaru,
*La classification des immersions combinatoires*, Inst. Hautes Études Sci. Publ. Math.**23**(1964), 75–91 (French). MR**172296**, DOI 10.1007/BF02684311 - Morris W. Hirsch,
*On the fibre homotopy type of normal bundles*, Michigan Math. J.**12**(1965), 225–229. MR**184251** - F. Hirzebruch and K. H. Mayer,
*$\textrm {O}(n)$-Mannigfaltigkeiten, exotische Sphären und Singularitäten*, Lecture Notes in Mathematics, No. 57, Springer-Verlag, Berlin-New York, 1968 (German). MR**0229251**, DOI 10.1007/BFb0074355 - Santiago López de Medrano,
*Some results on involutions of homotopy spheres*, Proc. Conf. on Transformation Groups (New Orleans, La., 1967) Springer, New York, 1968, pp. 167–174. MR**0253349**
J. Milnor, - John Milnor,
*Singular points of complex hypersurfaces*, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR**0239612** - Deane Montgomery and C. T. Yang,
*Free differentiable actions on homotopy spheres*, Proc. Conf. on Transformation Groups (New Orleans, La., 1967) Springer, New York, 1968, pp. 175–192. MR**0245042** - Claude Morlet,
*Les voisinages tubulaires des variétés semi-linéaires*, C. R. Acad. Sci. Paris Sér. A-B**262**(1966), A740–A743 (French). MR**214078** - Paul Olum,
*Mappings of manifolds and the notion of degree*, Ann. of Math. (2)**58**(1953), 458–480. MR**58212**, DOI 10.2307/1969748 - Elmer Rees,
*Embeddings of real projective spaces*, Topology**10**(1971), 309–312. MR**288778**, DOI 10.1016/0040-9383(71)90024-3 - C. P. Rourke and B. J. Sanderson,
*Block bundles. I*, Ann. of Math. (2)**87**(1968), 1–28. MR**226645**, DOI 10.2307/1970591 - N. E. Steenrod,
*Cohomology operations*, Annals of Mathematics Studies, No. 50, Princeton University Press, Princeton, N.J., 1962. Lectures by N. E. Steenrod written and revised by D. B. A. Epstein. MR**0145525**
D. Sullivan, - Hirosi Toda,
*Order of the identity class of a suspension space*, Ann. of Math. (2)**78**(1963), 300–325. MR**156347**, DOI 10.2307/1970345 - C. T. C. Wall,
*Surgery on compact manifolds*, London Mathematical Society Monographs, No. 1, Academic Press, London-New York, 1970. MR**0431216** - C. T. C. Wall,
*Free piecewise linear involutions on spheres*, Bull. Amer. Math. Soc.**74**(1968), 554–558. MR**222905**, DOI 10.1090/S0002-9904-1968-12006-3

*Microbundles and differentiable structures*, Princeton University, Princeton, N. J., 1961 (mimeographed notes).

*Triangulating homotopy equivalences*, Ph. D. Thesis, Princeton University, Princeton, N. J., 1966.

## Additional Information

- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**178**(1973), 193-225 - MSC: Primary 57D90; Secondary 57D65
- DOI: https://doi.org/10.1090/S0002-9947-1973-0324717-0
- MathSciNet review: 0324717