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Rational homotopy type and computability

Fedor Manin

TL;DR

It is shown that for a fixed Y, this question is algorithmically decidable for all X, A, and f if and only if Y has the rational homotopy type of an H-space.

Abstract

Given a simplicial pair $(X,A)$, a simplicial complex $Y$, and a map $f:A \to Y$, does $f$ have an extension to $X$? We show that for a fixed $Y$, this question is algorithmically decidable for all $X$, $A$, and $f$ if $Y$ has the rational homotopy type of an H-space. As a corollary, many questions related to bundle structures over a finite complex are likely decidable. Conversely, for all other $Y$, the question is at least as hard as certain special cases of Hilbert's tenth problem which are known or suspected to be undecidable.

Rational homotopy type and computability

TL;DR

It is shown that for a fixed Y, this question is algorithmically decidable for all X, A, and f if and only if Y has the rational homotopy type of an H-space.

Abstract

Given a simplicial pair , a simplicial complex , and a map , does have an extension to ? We show that for a fixed , this question is algorithmically decidable for all , , and if has the rational homotopy type of an H-space. As a corollary, many questions related to bundle structures over a finite complex are likely decidable. Conversely, for all other , the question is at least as hard as certain special cases of Hilbert's tenth problem which are known or suspected to be undecidable.

Paper Structure

This paper contains 20 sections, 23 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Examples
  3. Proof ideas
  4. Computational complexity
  5. The fiberwise case
  6. Structure of the paper
  7. Rational homotopy theory
  8. H-spaces
  9. The algebraic structure of $[X,Y]^f$
  10. An H-space action on $Y_n$
  11. The action of $[X/A,H_d]$ on $[X,Y_d]^f$
  12. The Postnikov induction
  13. Building blocks of homotopy-theoretic computation
  14. Computing $[X,Y]^f$
  15. Setup
  16. ...and 5 more sections

Key Result

Theorem A

Let $Y$ be a simply connected simplicial complex of finite type and $d \geq 2$, and suppose Then for any simplicial pair $(X,A)$ of cohomological dimension $d+1$ and simplicial map $f:A \to Y$, the existence of a continuous extension of $f$ to $X$ is decidable. Moreover, there is an algorithm which, given a simply connected simplicial complex $Y$, a simplicial pair $(X,A)$ of finite comple

Theorems & Definitions (45)

  • Theorem A
  • Example 1.3
  • Remark 1.4
  • Theorem 2.2: Sullivan's finiteness theorem SulLong
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 35 more