A CERTAIN PROPERTY OF POLYNOMIALS AND THE CI-STABILITY OF TANGENT BUNDLE OVER PROJECTIVE SPACES

Title & Authors
A CERTAIN PROPERTY OF POLYNOMIALS AND THE CI-STABILITY OF TANGENT BUNDLE OVER PROJECTIVE SPACES
Tanaka, Ryuichi;

Abstract
We determine the largest integer i such that $0 is odd in the polynomial $\small{(1+t+t^{2}+{\cdots}+t^{n})^{n+1}}$. We apply this to prove that the co-index of the tangent bundle over $\small{FP^{n}}$ is stable if$2^{r}{\leq}n<2^{r}+\frac{1}{3}(2^{r}-2)\$ for some integer r.
Keywords
sphere bundle;$\small{\mathbb{Z}_2-map}$;co-index;
Language
English
Cited by
References
1.
P. E. Conner and E. E. Floyd, Fixed point free involutions and equivariant maps, Bull. Amer. Math. Soc. 66 (1960), 416-441

2.
P. E. Conner and E. E. Floyd, Fixed point free involutions and equivariant maps II, Trans. Amer. Math. Soc. 105 (1962), 222-228

3.
A. Haefliger and M. W. Hirsch, Immersions in the stable range, Ann. of Math. (2) 75 (1962), 231-241

4.
R. Tanaka, On the index and co-index of sphere bundles, Kyushu J. Math. 57 (2003), no. 2, 371-382

5.
R. Tanaka, On the stability of (co-)index of sphere bundles, Kyushu J. Math. 59 (2005), no. 2, 321-331

6.
R. Tanaka, The index and co-index of the twisted tangent bundle over projective spaces, Math. J. Ibaraki Univ. 37 (2005), 35-38