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On the closed geodesics problem

Bitjong Ndombol

Abstract

Let $l\!k$ be a field of characteristic $p\geq 0$ and $X$ a simply connected finite CW complex. In this text, we prove that: {\sl if the cohomology algebra $H^*(X;l\!k)$ is generated, as an algebra, by at least two linearly independent elements, then the sequence of Betti numbers $ \left( \dim H^n(LX;l\!k)\right)_{n\geq 1 }$ grows unbounded.

On the closed geodesics problem

Abstract

Let be a field of characteristic and a simply connected finite CW complex. In this text, we prove that: {\sl if the cohomology algebra is generated, as an algebra, by at least two linearly independent elements, then the sequence of Betti numbers grows unbounded.
Paper Structure (23 sections, 25 theorems, 55 equations)

This paper contains 23 sections, 25 theorems, 55 equations.

Table of Contents

  1. Introduction.
  2. Theorem GM
  3. Main Theorem
  4. Corollary
  5. Theorem 1
  6. Theorem 2
  7. Algebraic preliminaries
  8. The Adams reduced bar construction
  9. Adams reduced cobar construction
  10. Further algebraic tools
  11. On minimal models
  12. Proof of Theorem 1
  13. First step
  14. Second step
  15. End of proof
  16. ...and 8 more sections

Key Result

Lemma 2.3.3

(S-E) Let $X$ be a topological space and $l\!k$ a commutative ring with unit. The DG algebra $(A, d_A) =C^*(X; l\!k)$ of normalized singular cochains on $X$ with coefficients in $l\!k$ has a cup-one product .

Theorems & Definitions (53)

  • Definition 2.3.1
  • Definition 2.3.2
  • Lemma 2.3.3
  • Remark 2.3.5
  • Definition 2.3.6
  • Lemma 2.3.7
  • proof
  • Lemma 2.3.8
  • proof
  • Remark 2.3.9
  • ...and 43 more