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Computing the Hopf invariant

Oleg R. Musin, Timur Shamazov

TL;DR

The paper develops a simplicial framework for computing the Hopf invariant of maps $S^{2n-1}\to S^n$ by translating Whitehead's integral approach into cochain language. It constructs a cochain $\theta$ satisfying $\delta\theta=f^*\omega$, and computes $H(f)$ as the pairing $\langle \theta \cup f^*\omega, \Delta \rangle$, enabling an explicit algorithm. The method reduces to solving a sparse linear system and summing over a fundamental cycle, with overall complexity $O(f_n^3)$; it handles odd $n$ trivially with zero invariant and asserts no superior local combinatorial formula exists. This provides a practical, general procedure for determining Hopf classes of simplicial maps.

Abstract

We consider Whitehead's integral formula and propose an algorithm for computing the Hopf invariant for simplicial mappings.

Computing the Hopf invariant

TL;DR

The paper develops a simplicial framework for computing the Hopf invariant of maps by translating Whitehead's integral approach into cochain language. It constructs a cochain satisfying , and computes as the pairing , enabling an explicit algorithm. The method reduces to solving a sparse linear system and summing over a fundamental cycle, with overall complexity ; it handles odd trivially with zero invariant and asserts no superior local combinatorial formula exists. This provides a practical, general procedure for determining Hopf classes of simplicial maps.

Abstract

We consider Whitehead's integral formula and propose an algorithm for computing the Hopf invariant for simplicial mappings.

Paper Structure

This paper contains 3 sections, 5 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Hopf invariant for simplicial maps
  3. Algorithm and its computational complexity for computing the Hopf invariant

Key Result

Lemma 1

The Hopf invariant $H(f)$ of $f$ is given by the pairing of the cochain $\theta \smile f^{*}\omega$ with the fundamental class $\Delta$ of $S^{2n-1}$:

Theorems & Definitions (7)

  • Lemma 1
  • Lemma 2
  • Theorem 3
  • proof
  • Theorem 4
  • Theorem 5
  • proof