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Every value of a moment map gives a smooth symplectic quotient

Statement

For every value of a moment map the level quotient is a smooth symplectic manifold. This is false.

Facts & Assumptions

Given: ${A C}_{ω}$ , $G = R$ acting on $M = R^{2}$ with $ω = d x \land d y$ by $t \cdot (x, y) = (e^{t} x, e^{- t} y)$ , its moment map, and the value $0$ .

[A1]

${A C}_{ω}$ is countable choice; it is used only through the fundamental-field interface.

[F1]

The fundamental field of $ξ = 1$ is $ξ_{M} = {\frac{d}{d t} ∣}_{0} (- t) \cdot (x, y) = - x \partial_{x} + y \partial_{y}$ , so $- ι_{ξ_{M}} ω = d (x y)$ : the map $μ (x, y) = x y$ satisfies the component equation. Fundamental vector fields for a left action, Moment map, component Hamiltonians and infinitesimal moment maps.

[F2]

The group is abelian, so the coadjoint action is trivial and equivariance of $μ$ amounts to invariance; $x y$ is invariant because $(e^{t} x) (e^{- t} y) = x y$ . Hence $μ$ is an equivariant moment map. Moment map, component Hamiltonians and infinitesimal moment maps.

[F3]

The reduction theorem requires a regular value and a free proper stabilizer action on the level; it is the only construction on this page that produces a smooth symplectic quotient. Marsden--Weinstein--Meyer symplectic reduction, Regularity of a moment map is equivalent to local freeness.

Refutation

technique · direct

1.1

By [F1] and [F2], $μ (x, y) = x y$ is an equivariant moment map for the action, and its value $0$ is attained exactly on the union $X = {x y = 0}$ of the two coordinate axes.

F1F2given

2.1

The value $0$ is critical: $d μ_{(0, 0)} = 0$ , so $0$ is not a regular value and [F3] does not apply to this value.

step 1.1F3

2.2

The orbits of the action inside $X$ are computed directly: the origin is a fixed point, and each of the four open half-axes is a single orbit, because $t \cdot (x_{0}, 0) = (e^{t} x_{0}, 0)$ runs through the half-axis as $t$ ranges over $R$ (and likewise on the $y$ -axis). Hence the quotient space $X / G$ has exactly five points.

step 1.1

3.1

In $X$ with the subspace topology, no neighbourhood of the origin is contained in ${0}$ : every ball around the origin meets the four half-axes away from the origin. Since the origin is a fixed point, its saturation is itself, so the class $[0]$ in $X / G$ is not an open point.

step 2.2

4.1

The quotient $X / G$ is therefore a five-point space with a non-open point. A smooth manifold containing a point with no open neighbourhood contained in that point cannot be zero-dimensional, since a zero-dimensional manifold is discrete; a positive-dimensional manifold has a neighbourhood homeomorphic to some $R^{n}$ with $n \geq 1$ , hence uncountably many points, which five points cannot supply. Thus $X / G$ is not a smooth manifold, and the value $0$ of the moment map does not produce a smooth symplectic quotient.

step 3.1A1∎

Depends on

Used by

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Dependency tree · two levels

31 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Eckhard Meinrenken, Symplectic Geometry (standard reference, not scraped)
Ana Cannas da Silva, Lectures on Symplectic Geometry (standard reference, not scraped)