Descending maps to quotient manifolds #
Let a group G act freely and properly discontinuously by C^n maps on a C^n manifold M. The
orbit space MulAction.orbitRel.Quotient G M is then a C^n manifold whose orbit projection is a
surjective C^n local diffeomorphism (TauCeti.isLocalDiffeomorph_quotientMk). A map f : M โ N
that is invariant under G descends to the orbit space
(TauCeti.MulAction.orbitRelQuotientLift). This file shows that the descended map is C^m for
m โค n exactly when f is (continuity is TauCeti.continuous_orbitRelQuotientLift_iff), and that
an equivariant C^m map between two such manifolds induces a C^m map of their orbit spaces
(MulActionHom.orbitRelQuotientMap). The same holds for submersions: an equivariant C^n
submersion with complement F induces a C^n submersion with complement F of orbit spaces,
because both orbit projections are local diffeomorphisms.
Over ๐ = โ with n = m = โ this is holomorphic descent: an invariant holomorphic map on a
complex manifold descends to a holomorphic map on the quotient by a free properly discontinuous
group of biholomorphisms. Conversely, a map on the quotient is the descent of its pullback along
the orbit projection (TauCeti.MulAction.eq_orbitRelQuotientLift_iff), so it is holomorphic
exactly when that pullback is. The hypotheses ask for an action by holomorphic maps; an action
that is only smooth after restricting scalars to โ gives the real statement instead.
Main results #
TauCeti.contMDiffAt_orbitRelQuotientLift_iffandTauCeti.contMDiff_orbitRelQuotientLift_iff: the descent of an invariant map isC^m, at the orbit of a point or globally, exactly when the map is.TauCeti.contMDiff_orbitRelQuotientMap: the map of orbit spaces induced by an equivariantC^mmap isC^m.TauCeti.isSubmersionOfComplement_orbitRelQuotientMap: the map of orbit spaces induced by an equivariantC^nsubmersion with complementFis aC^nsubmersion with complementF;TauCeti.isSubmersion_orbitRelQuotientMapis the version without a fixed complement.
References #
- John M. Lee, Introduction to Smooth Manifolds, second edition, Graduate Texts in Mathematics 218, Springer, 2013, Chapter 21 (quotients by discrete group actions).
- Otto Forster, Lectures on Riemann Surfaces, Graduate Texts in Mathematics 81, Springer, 1981, ยงยง4โ5 (holomorphic maps through coverings).
For a free properly discontinuous action by C^n maps on a C^n manifold and m โค n, the
descent of a G-invariant map is C^m at the orbit of x exactly when the map is C^m at
x.
For a free properly discontinuous
additive action by C^n maps and m โค n, the descent is C^m at the orbit of x exactly
when the original map is C^m at x.
For a free properly discontinuous action by C^n maps on a C^n manifold and m โค n, the
descent of a G-invariant map is C^m exactly when the map is. For holomorphic actions on complex
manifolds, this is the holomorphic descent of invariant holomorphic maps.
For a free properly discontinuous
additive action by C^n maps and m โค n, the descent is C^m exactly when the original map
is C^m.
For free properly discontinuous actions by C^n maps on C^n manifolds and m โค n, a
C^m map carrying one action to the other along ฯ induces a C^m map of orbit spaces.
For free properly discontinuous actions by C^n maps on C^n manifolds, a C^n submersion
with complement F carrying one action to the other along ฯ induces a C^n submersion with
complement F of orbit spaces. For holomorphic actions on complex manifolds, an equivariant
holomorphic submersion descends to a holomorphic submersion of the quotients.
For free properly discontinuous actions by C^n maps on C^n manifolds, a C^n submersion
carrying one action to the other along ฯ induces a C^n submersion of orbit spaces.