FoxDifferential.Discrete.Jacobian

3 sections | 3 files | 31 declarations

This module develops Fox differentials and completed Fox coordinates for free, profinite, and pro-\(C\) group constructions.

Automorphism

1 files | 7 declarations | 6 Theorem | 1 Definition

The Fox differential coordinate map is evaluated by the generator formula and the crossed-derivation rule.

Basic

1 files | 14 declarations | 10 Theorem | 4 Definition

The relative Fox Jacobian of \(\varphi : \mathrm{FreeGroup}(X) \to \mathrm{FreeGroup}(Y)\), with coefficients pushed forward by \(\psi : \mathrm{FreeGroup}(Y) \to H\). The row i...

ChainRule

1 files | 10 declarations | 10 Theorem

The composed derivative \(w \mapsto D(\varphi(w))\) is a crossed differential.