Difference between revisions of "Slice theorem"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (MR/ZBL numbers added) |
||
Line 1: | Line 1: | ||
− | A theorem reducing the description of the action of a transformation group on some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s1201501.png" /> of a given [[Orbit|orbit]] to that of the [[Stabilizer|stabilizer]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s1201502.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s1201503.png" /> of this orbit on some space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s1201504.png" /> which is | + | A theorem reducing the description of the action of a transformation group on some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s1201501.png" /> of a given [[Orbit|orbit]] to that of the [[Stabilizer|stabilizer]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s1201502.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s1201503.png" /> of this orbit on some space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s1201504.png" /> which is "normal" to the orbit at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s1201505.png" />. Namely, this theorem claims that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s1201506.png" /> is the homogeneous fibre space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s1201507.png" /> with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s1201508.png" />. Below, the precise setting and formulation are given, together with a counterpart of the theorem for algebraic transformation groups, called the étale slice theorem. |
==Slice theorem for topological transformation groups.== | ==Slice theorem for topological transformation groups.== | ||
Line 31: | Line 31: | ||
==Slice theorem for algebraic transformation groups.== | ==Slice theorem for algebraic transformation groups.== | ||
− | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150114.png" /> be an algebraic transformation group of an [[Algebraic variety|algebraic variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150115.png" />, all defined over an [[Algebraically closed field|algebraically closed field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150116.png" />. It happens very rarely that the literal analogue of the slice theorem above holds in this setting. Loosely speaking, the reason is that Zariski-open sets are | + | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150114.png" /> be an algebraic transformation group of an [[Algebraic variety|algebraic variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150115.png" />, all defined over an [[Algebraically closed field|algebraically closed field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150116.png" />. It happens very rarely that the literal analogue of the slice theorem above holds in this setting. Loosely speaking, the reason is that Zariski-open sets are "too big" (see [[#References|[a12]]], 6.1). One obtains an algebraic counterpart of a compact Lie group action on a differentiable manifold by taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150117.png" /> to be a [[Reductive group|reductive group]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150118.png" /> an [[Affine variety|affine variety]]. In this setting, a counterpart of a slice is given by the notion of an étale slice, defined as follows. |
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150119.png" /> be a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150120.png" /> such that the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150121.png" /> is closed. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150122.png" /> be an affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150123.png" />-invariant subvariety of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150124.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150125.png" />. As above, one can consider the homogeneous fibre space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150126.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150127.png" /> with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150128.png" />, and mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150129.png" />. In this situation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150130.png" /> is an affine variety, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150131.png" /> is a [[Morphism|morphism]], there are the categorical quotients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150132.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150133.png" /> and an induced morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150134.png" />, cf. [[#References|[a12]]]. The subvariety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150135.png" /> is called an étale slice at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150136.png" /> if | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150119.png" /> be a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150120.png" /> such that the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150121.png" /> is closed. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150122.png" /> be an affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150123.png" />-invariant subvariety of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150124.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150125.png" />. As above, one can consider the homogeneous fibre space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150126.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150127.png" /> with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150128.png" />, and mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150129.png" />. In this situation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150130.png" /> is an affine variety, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150131.png" /> is a [[Morphism|morphism]], there are the categorical quotients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150132.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150133.png" /> and an induced morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150134.png" />, cf. [[#References|[a12]]]. The subvariety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150135.png" /> is called an étale slice at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150136.png" /> if | ||
Line 46: | Line 46: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G.E. Bredon, "Introduction to compact transformation groups" , Acad. Press (1972) {{MR|0413144}} {{ZBL|0246.57017}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A.M. Gleason, "Spaces with a compact Lie group of transformations" ''Proc. Amer. Math. Soc.'' , '''1''' (1950) pp. 35–43 {{MR|0033830}} {{ZBL|0041.36207}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> K. Jänich, "Differenzierbare <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150152.png" />-Mannigfaltigkeiten" , ''Lecture Notes Math.'' , '''6''' , Springer (1968) {{MR|0202157}} {{ZBL|0153.53703}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J.L. Koszul, "Sur certains groupes de transformation de Lie" ''Colloq. Inst. C.N.R.S., Géom. Diff.'' , '''52''' (1953) pp. 137–142 {{MR|0059919}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> Koszul, J.L., "Lectures on groups of transformations" , Tata Inst. (1965) {{MR|218485}} {{ZBL|0195.04605}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> Luna,D., "Slices étales" ''Bull. Soc. Math. France'' , '''33''' (1973) pp. 81–105 {{MR|}} {{ZBL|0286.14014}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> Luna, D., "Sur certaines opérations différentiables des groups de Lie" ''Amer. J. Math.'' , '''97''' (1975) pp. 172–181</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> D. Montgomery, C.T. Yang, "The existence of slice" ''Ann. of Math.'' , '''65''' (1957) pp. 108–116 {{MR|}} {{ZBL|0078.16202}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> G.D. Mostow, "On a theorem of Montgomery" ''Ann. of Math.'' , '''65''' (1957) pp. 432–446</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> R. Palais, "Embeddings of compact differentiable transformation groups in orthogonal representations" ''J. Math. Mech.'' , '''6''' (1957) pp. 673–678</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> R.S. Palais, "Slices and equivariant imbeddings" , ''Sem. Transformation Groups'' , Princeton Univ. Press (1960)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> V.L. Popov, E.B. Vinberg, "Invariant theory" , ''Algebraic Geometry IV'' , ''Encycl. Math. Sci.'' , '''55''' , Springer (1994) pp. 122–284 {{MR|1456471}} {{ZBL|1099.13012}} {{ZBL|1088.81075}} {{ZBL|1065.82003}} {{ZBL|1053.82006}} {{ZBL|0783.14028}} {{ZBL|0754.13005}} {{ZBL|0736.15019}} {{ZBL|0735.14010}} {{ZBL|0789.14008}} {{ZBL|0679.14024}} {{ZBL|0491.14004}} {{ZBL|0478.14006}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> Wu Yi Hsiang, "Cohomology theory of topological transformation groups" , ''Ergebn. Math.'' , '''85''' , Springer (1979) {{MR|}} {{ZBL|0511.57002}} </TD></TR></table> |
Revision as of 21:56, 30 March 2012
A theorem reducing the description of the action of a transformation group on some neighbourhood of a given orbit to that of the stabilizer
of a point
of this orbit on some space
which is "normal" to the orbit at
. Namely, this theorem claims that
is the homogeneous fibre space over
with fibre
. Below, the precise setting and formulation are given, together with a counterpart of the theorem for algebraic transformation groups, called the étale slice theorem.
Slice theorem for topological transformation groups.
Let be a topological transformation group of a Hausdorff space
. A subspace
of
is called a slice at a point
if the following conditions hold:
i) is invariant under the stabilizer
of
;
ii) the union of all orbits intersecting
is an open neighbourhood of the orbit
of
;
iii) if is the homogeneous fibre space over
with fibre
, then the equivariant mapping
, which is uniquely defined by the condition that its restriction to the fibre
over
is the identity mapping
(cf. also Equivariant cohomology), is a homeomorphism of
onto
. Equivalent definitions are obtained by replacing iii) either by:
iv) there is an equivariant mapping that is the identity on
and is such that
; or by
v) is closed in
and
implies
.
The slice theorem claims that if certain conditions hold, then there is a slice at a point . The conditions depend on the case under consideration. The necessity of certain conditions is explained by the following observation.
If there is a slice at
, then there is a neighbourhood
of
(namely,
) such that the stabilizer of every point of
is conjugate to a subgroup of
. In general, this property fails (e.g., take
acting on the space of binary forms of degree
in the variables
,
by linear substitutions. Then the stabilizer of
is trivial but every neighbourhood of
contains a point whose stabilizer has order
.)
The first case in which the validity of the slice theorem was investigated is that of a compact Lie group . In this case, it has been proved that if
is a fully regular space, then there is a slice at every point
. If, moreover,
is a differentiable manifold and
acts smoothly, then at every
there is a differentiable slice of a special kind. Namely, in this case there is an equivariant diffeomorphism
, being the identity on
, of the normal vector bundle (cf. also Normal space (to a surface)) of
onto an open neighbourhood of
in
. The image under
of the fibre of this bundle over
is a slice at
which is a smooth submanifold of
diffeomorphic to a vector space.
A particular case of the slice theorem for compact Lie groups was for the first time ever proven in [a2]. Then the differentiable and general versions, formulated above, were proven, respectively in [a4] and [a8], [a9], [a10].
The slice theorem is an indispensable tool in the theory of transformation groups, which frequently makes it possible to reduce an investigation to simple group actions like linear ones. In particular, the slice theorem is the key ingredient in the proofs of the following two basic facts of the theory:
1) If is a compact Lie group,
a separable metrizable space and there are only finitely many conjugacy classes of stabilizers of points in
, then there is an equivariant embedding of
in a Euclidean vector space endowed with an orthogonal action of
.
2) Let be a compact Lie group acting smoothly on a connected differentiable manifold
. Then there are a subgroup
of
and a dense open subset
of
such that
and the stabilizer of every point
is conjugate to a subgroup of
which coincides with
if
.
There are versions of the slice theorem for non-compact groups. For instance, let be an algebraic complex reductive group and
its finite-dimensional algebraic representation, both defined over the real numbers
. Let
be the Lie group of real points of
and
a subgroup of
containing the connected component of identity element. Let
be a closed
-invariant differentiable submanifold of
, the space of real points of
. Then, [a7], for every closed orbit
in
there is an equivariant diffeomorphism
of a
-invariant neighbourhood
of
in the normal vector bundle of
onto a
-invariant saturated neighbourhood
of
in
(a neighbourhood
is saturated if the fact fact that the closure of an orbit intersects
implies that this orbit lies in
). In this case, the image under
of the fibre of the natural projection
over
is a slice at
for the action of
on
.
Slice theorem for algebraic transformation groups.
Let be an algebraic transformation group of an algebraic variety
, all defined over an algebraically closed field
. It happens very rarely that the literal analogue of the slice theorem above holds in this setting. Loosely speaking, the reason is that Zariski-open sets are "too big" (see [a12], 6.1). One obtains an algebraic counterpart of a compact Lie group action on a differentiable manifold by taking
to be a reductive group and
an affine variety. In this setting, a counterpart of a slice is given by the notion of an étale slice, defined as follows.
Let be a point of
such that the orbit
is closed. Let
be an affine
-invariant subvariety of
containing
. As above, one can consider the homogeneous fibre space
over
with fibre
, and mapping
. In this situation,
is an affine variety,
is a morphism, there are the categorical quotients
,
and an induced morphism
, cf. [a12]. The subvariety
is called an étale slice at
if
i) is obtained from
by means of the base change
; and
ii) is an étale morphism.
The étale slice theorem, proved in [a6], claims that there is an étale slice at every point such that the orbit
is closed.
If is
, the field of complex numbers, and
is a smooth point of
, then the étale slice theorem implies that there exists an analytic slice at
. More precisely, there is an invariant analytic neighbourhood of
in
which is analytically isomorphic to an invariant analytic neighbourhood of
in the normal vector bundle of
, cf. [a7], [a12].
Like the slice theorem for topological transformation groups, the étale slice theorem is an indispensable result in the investigation of algebraic transformation groups; see [a6] for some basic results deduced from this theorem.
References
[a1] | G.E. Bredon, "Introduction to compact transformation groups" , Acad. Press (1972) MR0413144 Zbl 0246.57017 |
[a2] | A.M. Gleason, "Spaces with a compact Lie group of transformations" Proc. Amer. Math. Soc. , 1 (1950) pp. 35–43 MR0033830 Zbl 0041.36207 |
[a3] | K. Jänich, "Differenzierbare ![]() |
[a4] | J.L. Koszul, "Sur certains groupes de transformation de Lie" Colloq. Inst. C.N.R.S., Géom. Diff. , 52 (1953) pp. 137–142 MR0059919 |
[a5] | Koszul, J.L., "Lectures on groups of transformations" , Tata Inst. (1965) MR218485 Zbl 0195.04605 |
[a6] | Luna,D., "Slices étales" Bull. Soc. Math. France , 33 (1973) pp. 81–105 Zbl 0286.14014 |
[a7] | Luna, D., "Sur certaines opérations différentiables des groups de Lie" Amer. J. Math. , 97 (1975) pp. 172–181 |
[a8] | D. Montgomery, C.T. Yang, "The existence of slice" Ann. of Math. , 65 (1957) pp. 108–116 Zbl 0078.16202 |
[a9] | G.D. Mostow, "On a theorem of Montgomery" Ann. of Math. , 65 (1957) pp. 432–446 |
[a10] | R. Palais, "Embeddings of compact differentiable transformation groups in orthogonal representations" J. Math. Mech. , 6 (1957) pp. 673–678 |
[a11] | R.S. Palais, "Slices and equivariant imbeddings" , Sem. Transformation Groups , Princeton Univ. Press (1960) |
[a12] | V.L. Popov, E.B. Vinberg, "Invariant theory" , Algebraic Geometry IV , Encycl. Math. Sci. , 55 , Springer (1994) pp. 122–284 MR1456471 Zbl 1099.13012 Zbl 1088.81075 Zbl 1065.82003 Zbl 1053.82006 Zbl 0783.14028 Zbl 0754.13005 Zbl 0736.15019 Zbl 0735.14010 Zbl 0789.14008 Zbl 0679.14024 Zbl 0491.14004 Zbl 0478.14006 |
[a13] | Wu Yi Hsiang, "Cohomology theory of topological transformation groups" , Ergebn. Math. , 85 , Springer (1979) Zbl 0511.57002 |
Slice theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Slice_theorem&oldid=13508