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The index of certain Stiefel manifolds

Samik Basu, Bikramjit Kundu

Abstract

This paper computes the Fadell-Husseini index of Stiefel manifolds in the case where the group acts via permutations of the orthogonal vectors. The computations are carried out in the case of elementary Abelian $p$-groups. The results are shown to imply certain generalizations of the Kakutani-Yamabe-Yujobo theorem.

The index of certain Stiefel manifolds

Abstract

This paper computes the Fadell-Husseini index of Stiefel manifolds in the case where the group acts via permutations of the orthogonal vectors. The computations are carried out in the case of elementary Abelian -groups. The results are shown to imply certain generalizations of the Kakutani-Yamabe-Yujobo theorem.

Paper Structure

This paper contains 8 sections, 23 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Organization
  3. Preliminaries
  4. Homotopy orbits of Stiefel manifolds
  5. Index computations
  6. Computations at odd primes
  7. Computations at $p=2$
  8. Applications

Key Result

Proposition 2.3

Let $X$ be a finite dimensional $C_p^n$-CW-complex. The fixed point space $X^{C_p^n}\neq \varnothing$$\iff$$\hbox{Index}_{C_p^n}(X)= 0$.

Theorems & Definitions (40)

  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Theorem 2.7
  • Definition 2.9
  • Proposition 2.10
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • ...and 30 more