nLab jet group

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

(see Kolář, Michor & Slovák 1993, §13.1 & Prop. on p. 131; Dartnell 1994, section 1, also Grasseau 2006, pp. 14).

Group homology

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

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

(Dartnell 94, theorem 1.1)

Related concepts

• higher order frame bundle

• jet groupoid

References

Original discussion in the context of integrability of G-structures

• Victor Guillemin, section 3 of: The integrability problem for $G$-structures, Trans. Amer. Math. Soc. 116 (1965), 544-560 [jstor:1994134]

and in the context of natural vector bundles:

• Chuu-Lian Terng, p. 779 of: Natural vector bundles and natural differential operators, Amer. J. Math. 100 (1978) 775-828 [doi:10.2307/2373910, jstor:2373910]

Textbook accounts and lecture notes:

• Demeter Krupka, Josef Janyška, §2.2 in: Lectures on differential invariants, Univerzita J. E. Purkyně, Brno (1990) [ISBN:80-210-165-8, researchgate]

  (there jet groups are called “differential groups”)

• Ivan Kolář, Peter Michor, Jan Slovák, section 13 of: Natural operators in differential geometry, Springer (1993) [doi:10.1007/978-3-662-02950-3, book webpage, pdf]

See also

• Wikipedia, Jet group

• MO discussion

Discussion of the group homology of jet groups:

• Pablo Dartnell, On the homology of groups of jets, Journal of Pure and Applied Algebra 92 2 (1994) 109-121 [doi:10.1016/0022-4049(94)90017-5]

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

Discussion of jet-frame bundles and their Cartan geometry:

• Michael Grasseau, Jets, frames, and their Cartan geometry [arXiv:math-ph/0603063, hal:00021276]