# nLab jet group

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

group theory

# Contents

## Idea

The concept of jet group is the generalization of general linear group from first order to higher order jets.

In terms of synthetic differential geometry/differential cohesion a general linear group is the automorphism group of a first-order infinitesimal disk, while a jet group is the automorphism group of a higher order infinitesimal disk. See also at differential cohesion – Frame bundles.

## Properties

### Homotopy type

For all $k \in \mathbb{N}$, the homotopy type of the orientation preserving jet group $GL^k_p(n)$ is that of the ordinary orientation-preserving general linear group $GL(n)$, and the canonical projection

$GL^k_+(n) \longrightarrow GL_+(n)$

is, on the level of the underlying topological spaces, a homotopy equivalence, indeed it preserves the maximal compact subgroup, which is the special orthogonal group $SO(n)$ on both sides (recalled e.g. in Dartnell 94, section 1).

### Group homology

The canonical projection $GL^k_+(n) \longrightarrow GL_+(n)$ also induces an isomorphism on group homology with constant integer coefficients

$H_\bullet^{grp}(GL^k_+(n), \mathbb{Z}) \stackrel{\simeq}{\longrightarrow} H_\bullet^{grp}(GL_+(n),\mathbb{Z}) \,.$

## References

Original discussion (in the context of integrability of G-structures) is due to

• Victor Guillemin, section 3 of The integrability problem for $G$-structures, Trans. Amer. Math. Soc. 116 (1965), 544–560. (JSTOR)

Textbook accounts and lecture notes include

• C.L. Terng, Natural vector bundles and natural differential operators, Amer. J. Math. 100 (1978) 775-828.

• Demeter Krupka, Josef Janyška, Lectures on differential invariants, Univerzita JEP, Brno, 1990.