nLab
jet group

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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 kk \in \mathbb{N}, the homotopy type of the orientation preserving jet group GL p k(n)GL^k_p(n) is that of the ordinary orientation-preserving general linear group GL(n)GL(n), and the canonical projection

GL + k(n)GL +(n) 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)SO(n) on both sides (recalled e.g. in Dartnell 94, section 1).

Group homology

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

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

(Dartnell 94, theorem 1.1)

References

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

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

Textbook accounts and lecture notes include

See also

Discussion of the group homology of jet groups includes

  • Pablo Dartnell, On the homology of groups of jets, Journal of Pure and Applied Algebra Volume 92, Issue 2, 7 March 1994, Pages 109–121 (publisher)

  • Dror Farjoun, Jekel, Suciu, Homology of jet groups (pdf)

Last revised on January 16, 2015 at 20:09:28. See the history of this page for a list of all contributions to it.