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 (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

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

(Dartnell 94, theorem 1.1)

Related concepts

• higher order frame bundle

• jet groupoid

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.

• Ivan Kolář, Peter Michor, Jan Slovák, section 13 of Natural operators in differential geometry (pdf)

See also

• Wikipedia, Jet group

• MO discussion

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)