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A localization formula for equivariant
Lyusternik-Schnirelmann category
Marine Fontaine and James Montaldi
Abstract
The LS-category of a topological space is a numerical homotopy invari-
ant, introduced originally in a course on the global calculus of variations
by Lyusternik and Schnirelmann, to estimate the number of critical points
of a smooth function. When the topological space is a smooth manifold
equipped with a proper action of a Lie group, we give a localization for-
mula to calculate the equivariant analogue of this category in terms of the
minimal orbit-type strata. The formula holds provided that the manifold
admits a specific cover. We show that such a cover exists on every sym-
plectic toric manifold. The known result stating that the LS-category of a
symplectic toric manifold is equal to the number of fixed points of the torus
action follows from our localization formula.
Contents
1 Introduction 2
2 Terminologies 3
3 Stratifications and orbit-type strata 8
4 G-tubular covers 14
5 Localization Formula 17
6 Tubular covers of symplectic toric manifolds 21
1
arXiv:1712.07096v1 [math.GT] 19 Dec 2017
1. Introduction
The Lyusternik-Schnirelmann category or LS-category of a topological space M
is the numerical homotopy invariant Cat(M)defined to be the least number of
open subsets U⊂M, whose inclusion is nullhomotopic, that are required to cover
M. Although it is now the subject of a full theory in connection with algebraic
topology, it was originally introduced by Lyusternik and Schnirelmann in a course
on the global calculus of variations, when Mis a smooth closed manifold without
boundary [16]. In this case they show that any smooth real-valued function f
defined on Mhas at least Cat(M)critical points. The difference with Morse
theory is that fis allowed to have degenerate critical points. Rewiews on the
Lyusternik-Schnirelmann theory are for instance [14,6,3].
If Mis a topological space, acted on continuously and properly by a topological
group G, then the LS-category has an equivariant analogue CatG(M). It has first
been introduced by Fadell [9] and Marzantowicz [17] for compact groups,
and by Colman [5] for finite groups. A substantial part of the theory has been
extended for proper group actions by Ayala, Lasheras and Quintero [2].
AG-categorical open subset of Mis a G-invariant open subset U⊂Madmit-
ting a G-deformation retract onto a G-orbit (cf. Definition 2.1). The numerical ho-
motopy invariant CatG(M)is the least number (possibly infinite) of G-categorical
open subsets that are required to cover M. If Mis a smooth manifold and Gis a
Lie group, a class of G-categorical open subsets consists of G-tubular open subsets,
which are essentially tubular neighbourhoods of group orbits (cf. Definition 4.1).
This fact is a direct application of the Tube Theorem 2.1.
This invariant is in general difficult to compute and we are usually only able
to know an estimation of it, in term of the cup length of M. We obtain a new
formula to reduce the calculation of CatG(M)to the calculation of the equivariant
LS-category of the minimal orbit-type strata of M. In general, any topological
space Mcan be written as a disjoint union of smaller subsets Mβ, called strata,
indexed on some strictly partially ordered set (B,≺). Those strata are required
to fit in a specific way and form themselves a strictly partially ordered set (cf.
Section 3). A stratum is minimal if it is minimal with respect to the strict partial
order defined on them. If Mis a proper G-manifold, the strata Mβare generally
the connected components of the orbit-type submanifolds. We use a modified
2
definition of orbit-type stratum (cf. Definition 3.3). We say that an orbit-type
stratum is a G-orbit of a connected component of the subset of Mof all the points
having the same stabilizer.
On a large class of proper G-manifolds M, including symplectic toric manifolds,
we observe that Mcan be entirely covered by a subcover of its minimal orbit-type
strata, made of G-tubular open subsets. Besides this cover is the smallest cover,
made of G-categorical open subsets, that we can take. Such covers are called
minimal G-tubular covers and are discussed in Section 4. However those covers
do not exist in general. A non-example is given, when Mis a non-Hamiltonian
compact S1-manifold (cf. Example 4.3). By using the natural stratification of
the moment polytope, we show in Section 6that every symplectic toric manifold
admits a minimal G-tubular cover, where Gin this case is a torus having half the
dimension of Mand acting effectively on it (cf. Theorem 6.1). When Madmits a
minimal G-tubular cover, we show that the calculation of CatG(M)is intrinsically
reduced to those of the minimal orbit-type strata of M. Explicitly, we obtain the
localization formula
CatG(M) = XCatG(Mβ)
where the summation is taken over the minimal orbit-type strata Mβ(cf. Section
5, Theorem 5.1 and Corollary 5.2). The result of Bayeh and Sarkar (cf. [4]
Theorem 5.1), which states that the equivariant Lyusternik-Schnirelmann cate-
gory of a quasitoric manifold is precisely the number of fixed points of the torus
action, is a consequence of Theorem 6.1 and of our localization formula.
Acknowledgements. We would like to thank Yael Karshon and Eckhard Mein-
renken for useful discussions and suggestions.
After completing this work, which forms part of the thesis [10], we discoverd
the paper of Hurder and Töben [13] which contains one of our main theorems
(Theorem 5.1) and uses a similar approach.
2. Terminologies
We work with smooth manifolds and, except stated otherwise, the term subman-
ifold refers to an embedded submanifold. A smooth action of a Lie group G
3
on Mis a group homomorphism G→Diff(M). We denote the action map by
(g, m)∈G×M7→ g·m∈M. A G-manifold is a pair (M , G)where Mis a
smooth manifold acted on by a Lie group G. If the action is proper then we
refer to (M, G)as a proper G-manifold. A smooth map f:M→Nbetween two
G-manifolds is G-equivariant if f(g·m) = g·f(m)for all g∈Gand m∈M. The
stabilizer of m∈Mis the subgroup Gm={g∈G|g·m=m}. We say that the
action of Gon Mis free if all the stabilizers Gmare equal to the trivial group .
The group orbit or G-orbit of a point m∈Mis the set G·m={g·m|g∈G}.
We use the notation g·mto mean the tangent space to G·mat m. The space
gstands for the Lie algebra of Gwith Lie bracket [·,·], obtained by identifying g
with the left invariant vector fields on G.
Local models of G-orbits
Let (M, G)be a proper G-manifold. In this case, the group orbits are embedded
submanifolds and all the stabilizers Gmare compact. Let N⊂TmMbe a Gm-
invariant complement to g·min TmM. Given a subgroup Kof G, there is a (left)
K-action on the product G×Ngiven by
k·(g, ν) = (gk−1, k ·ν).(1)
This action is free and proper by freeness and properness of the action on the
first factor. The orbit space G×KNis thus a smooth manifold whose points are
equivalence classes of the form [(g, ν)], and the orbit map ρ:G×N→G×KNis a
smooth surjective submersion. Moreover the group Gacts smoothly and properly
on G×KN, by left multiplication on the first factor. The theorem below states
that, given a Gm-invariant neighbourhood of zero N0⊂N, the associated bundle
G×KN0defines a local model for some G-invariant open subset U⊂M.
Theorem 2.1 (Tube Theorem (cf. [18] Theorem 2.3.28)).Let (M, G)be
a proper G-manifold and set K=Gmfor some m∈M. Let N0⊂Nbe
an open K-invariant neighbourhood of 0. Then, there exists a G-invariant
neighbourhood U⊂Mof mand a G-equivariant diffeomorphism
ϕ:G×KN0→U(2)
4
sending [(e, 0)] on m.
G-categorical open subsets
Although the LS-category is defined for topological spaces, our interest is mainly
about proper G-manifolds (M , G). The terminologies are thus defined in this
setting. Given (M, G), a homotopy H:M×[0,1] →Mwhich satisfies H(g·m, t) =
g·H(m, t)for every g∈G,m∈Mand t∈[0,1] is called a G-homotopy. We write
Ht(m) = H(m, t). Let B⊂Abe two G-invariant subsets of M. A G-deformation
retract of Aonto Bis a G-homotopy H:A×[0,1] →Asuch that H0(a) = aand
H1(a)∈Bfor every a∈A, and H(b, 1) = bfor every b∈B.
Definition 2.1.AG-invariant subset U⊂Mis called G-categorical if there exists
aG-deformation retract of Uonto the orbit G·xof some x∈U.
Definition 2.2.The equivariant LS-category of M, denoted CatG(M), is the least
number of G-categorical open subsets U⊂Mthat are required to cover M. We
set CatG(M) = ∞if such a cover does not exist. The non-equivariant LS-category
Cat(M)is obtained by setting G=.
Example 2.3.The equivariant version of the LS-category is in general different
from its non-equivariant analogue, as shown in the examples below.
(i) Let M=S1×Rwith cylindrical coordinates (θ, z). Define an S1-action
on Mby φ·(θ, z) = (θ+φ, z). The cylinder itself is an S1-categorical
open subset with S1-deformation retract H:M×[0,1] →Mgiven by
H((θ, z), t)=(θ, (1 −t)z). Therefore, CatS1(M)=1. However we require
two contractible open subsets to cover M, which yields
1 = CatS1(M)<Cat(M)=2.
(ii) Consider the complex projective space M=CP2endowed with the S1-
action θ·[z0:z1:z2] = [eiθz0:z1:z2].For i= 0,1,2, the open subsets
Ui={[z0:z1:z2]|zi6= 0}are S1-invariant. On U0, an S1-deformation
retract onto an orbit is given by
H([z0:z1:z2], t)=[z0: (1 −t)z1: (1 −t)z2].
5
The image H1(U0)is the single point [1 : 0 : 0] which is a fixed point of the
action, hence an S1-orbit. Similar homotopies can be found on U1and U2,
respectively. Therefore CatS1(M)is at most three. The fact that we have
an equality follows from Proposition 4.1 below. We conclude that
CatS1(M) = Cat(M)=3.
(iii) The rotations of a tetrahedron form a group Tof order 12, which is a zero-
dimensional Lie subgroup of SO(3). This group acts on M=S2. We
construct a cover of Mby three T-categorical open subsets as follows:
Pick a point x1∈Mand its opposite point y1∈M. The T-orbit of x1forms
a spherical tetrahedron with vertices x1, x2, x3, x4. Similarly the T-orbit of
y1forms another spherical tetrahedron with vertices y1, y2, y3, y4. For each
i<jdenote by pij the middle point of the geodesic arc joining xiand xj
(cf. Figure 1).
Figure 1: Spherical tetrahedrons on the sphere.
For each i, let Di⊂Mbe an open disk centered at xisuch that
Di∩Dj={pij } ∀i < j.
In the same way, let for each i, an open disk Ei⊂Mcentered at yiwith
the property
∀i < j Ei∩Ej={pkl}where k, l /∈ {i, j}k < l
as shown in Figure 2.
6
Figure 2: Disk E4centered at y4.
Finally we define for each i < j, an open subset Bij ⊂Mcontaining pij
such that
xk∈Bij \Bij ∀k=i, j
yk∈Bij \Bij ∀k6=i, j
We obtain the following T-categorical open subsets:
D=
4
[
i=1
Di
which retracts in a T-equivariant way onto the orbit T·x1,
E=
4
[
i=1
Ei
which retracts in a T-equivariant way onto the orbit T·y1,and
B=[
i<j
Bij
which retracts in a T-equivariant way onto the orbit T·p12.Those three
subsets form a cover of M. This cover is in fact the smallest that we can
take, by Proposition 4.1 below. Hence
3 = CatT(M)>Cat(M)=2.
7
3. Stratifications and orbit-type strata
Apartition of a topological space Mis a cover of Mby pairwise disjoint subsets.
Clearly every topological space admits a partition into its connected components.
If our topological space is endowed with a group action, we can choose a partition
which also encodes the information about the group action. For example, a proper
G-manifold (M, G)can be partitioned into locally closed (connected) submanifolds
called the orbit-type strata, each of them being a union of group orbits with the
same orbit-type.
Stratifications
There are several ways to define stratifications. The one we present here is the
definition used by Kirwan in her thesis [15]. It is more flexible than the standard
definition of Duistermaat and Kolk [8] (Definition 2.7.3), especially for ap-
plications to algebraic geometry. A strict partial order ≺on Bis a binary relation
which is irreflexive and transitive. Note that in this case if α, β ∈ B are such that
α≺β, then β6≺ α. For example the set of conjugacy classes of subgroups of G
admits the strict partial order ≺conj , where we say that (K)≺conj (H)if and only
if His conjugate to a proper subgroup of K.
Example 3.1.The group Thas four conjugacy classes, namely (T),(Z3),(Z2)and
( ). There are partially ordered with respect to ≺conj as shown in Figure 3.
T
Z3Z2
Figure 3: Conjugacy classes of subgroups of Twhere
the order goes up to down i.e. (T)is minimal with
respect to ≺conj .
Definition 3.2.A collection {Mβ|β∈ B} of subsets of a topological space Mis
locally finite if each compact subset of Mmeets only finitely many Mβ. A locally
8
finite collection {Mβ|β∈ B} of locally closed (non-empty) topological subspaces
of Mform a B-decomposition or stratification of Mif Mis the disjoint union of
the strata Mβ, and there is a strict partial order ≺on the indexing set Bsuch
that
Mβ⊂[
αβ
Mα(3)
for every β∈ B. We say that the stratification is smooth if Mis a smooth manifold
and every Mβis a locally closed submanifold.
Given a stratification of M, a strict partial order can be defined on the strata
as follows
Mα< Mβ⇐⇒ α≺β. (4)
We say that a stratum Mβis minimal with respect to (4) if there is no α∈ B
such that Mα< Mβ. Of course minimal strata are not unique because we just
have a partial ordering. If Mβis a minimal stratum, (3) implies that Mβ⊂Mβ.
In particular Mβ=Mβi.e. Mβis closed in M.
Orbit-type strata
Let Gbe a Lie group and H⊂Gbe a closed subgroup. The conjugacy class of
His the set (H) = {L⊂G|L=gHg−1for some g∈G}.Given a G-manifold
(M, G), we define the set
M(H):= {m∈M|Gm∈(H)}
which is the union of all the G-orbits in Mwith orbit-type (H). Using the
definitions and the G-invariance of M(H), it is shown in [18] (Proposition 2.4.4)
that M(H)=G·MHwhere
MH={m∈M|Gm=H}.
Note that the biggest subgroup of Gwhich leaves MHinvariant is the normalizer
NG(H) = {g∈G|gHg−1=H}.Furthermore this action induces a well-defined
free action of the quotient group NG(H)/H on MH. Write
MH=a
b∈BH
MH,b
9
as the disjoint union of its connected components, indexed on some set BH. Given
b∈ BH, we define the equivalence class (b)to be the set of indices a∈ BHsuch
that G·MH,a =G·MH,b. Let Bbe the set of pairs β= ((H),(b)) where (H)is
the conjugacy class of some closed subgroup of Gand b∈ BH.
Definition 3.3.For β= ((H),(b)) ∈ B, we define an orbit-type stratum Mβto be
the G-orbit of the connected component MH,b of MH.
We use here a modified definition of the standard definition which states that
an orbit-type stratum is a connected component of M(H). If the G-action on Mis
proper, the connected components MH,b are locally closed embedded submanifolds
of Mand so are their G-orbits (cf. [18] Proposition 2.4.7).
The example below illustrates the difference between the standard definition
of orbit-type strata and ours. With our definition, the orbit-type strata might not
be connected.
Example 3.4.Think of R∗=R\ {0}as a multiplicative group and let it act on
M=R2by t·(x, y)=(x, ty). The stabilizers of points of Mare either equal
to the trivial group , or equal to R∗. Then M(R∗)=MR∗is the x-axis, and
M( ) =M=H+∪H−where H±={(x, y)∈M| ±y > 0}. According to the
standard definition of orbit-type strata, there are two strata with orbit-type ( ),
namely the connected components H+and H−; and one stratum with orbit-type
(R∗), the x-axis.
With our definition, there is one stratum with orbit-type (R∗)which is the
x-axis; but there is only one stratum with orbit-type ( ) which is H+∪H−.
Indeed, Mhas two connected components, H+and H−. The R∗-orbits of each
of them coincide. There is thus only one stratum with orbit-type ( ) and it is not
connected.
We define a strict partial order ≺on Bas follows: for α= ((K),(a)) and
β= ((H),(b)),
α≺β⇐⇒ α6=βand Mα∩Mβ6=∅.(5)
By α6=βwe mean that the associated orbit-type strata Mαand Mβare distinct.
10
Proposition 3.1 ( Sjamaar and Lerman [19]).Let (M, G)be a proper
G-manifold and let (B,≺)as above with partial order (5). Then the orbit-type
strata {Mβ|β∈ B} form a smooth stratification of M.
Note. In [19], Sjamaar and Lerman use the standard definition of orbit-type
strata. In particular those are connected submanifolds of M. In [10] we give a
proof using the modified definition of orbit-type strata (cf. [10] Proposition 2.2.9).
Example 3.5.Given an equivalence class (H), the corresponding orbit-type strata
might not all have the same dimension, as shown in the following example, ap-
pearing in Delzant [7] and Sjamaar and Lerman [19].
(i) Let M=CP2endowed with the S1-action
θ·[z0:z1:z2] = [eiθz0:z1:z2].
The set MS1has two connected components namely, the point [1 : 0 : 0]
and a copy of CP1, which consists of the points of the form [0 : z1:z2].
Since S1acts trivially on each of these components, they form themselves
two orbit-type strata, which are closed submanifolds of M. Since the action
is free anywhere else, the last orbit-type stratum is M\({[1 : 0 : 0]} ∪ CP1).
It has orbit-type ( ) and is an open dense submanifold of M.
(ii) Let M=S2be the 2-sphere embedded in R3, equipped with the S1-action
which rotates the sphere about the z-axis. There are three orbit-type strata
namely, M( ) which is diffeomorphic to S1×(−1,1) and the two closed
connected components of MS1that are the North and South pole.
(iii) The group Tacts on M=S2. This group contains a copy of the cyclic
group of order three C3'Z3for each vertex, one copy of Z2for each axis
joining the middle point of an edge and the middle point of the opposite
edge, and the identity (cf. Figure 4).
There are two minimal strata with orbit-type (Z3), one minimal stratum
with orbit-type (Z2), and one open dense stratum with orbit type ( ) (cf.
11
Figure 4: On the left hand side we fix a vertex vand
permute the three other vertices. As a subgroup it is
isomorphic to C3. On the right hand side we permute
v1, v2and v3, v4. This subgroup is isomorphic to Z2.
Figure 5). Indeed, when H=Z3, the eight points forming M(H)are a union
of two T-orbits. There are thus two strata with orbit-type (Z3). For H=Z2,
the six points forming M(H)are a single T-orbit and form a single stratum.
H MHM(H)
T ∅ ∅
Z3
Z2
M\ {14 points }M\ {14 points }
Figure 5: Orbit-type strata of (M, )where M=S2.
12
In general, using the strict partial order ≺conj on the conjugacy classes of
subgroups of Gis not enough to guarantee that we have a good stratification. For
instance, in Example 3.5 (iii), we have (T)≺conj (Z2)but there are no strata with
orbit-type (T). However we have the following lemma:
Lemma 3.2.If α= ((K),(a)) and β= ((H),(b)) then
α≺β=⇒(K)≺conj (H).
Proof — By definition α≺βimplies that there exists some x∈Mα∩Mβ. In
particular x∈Mαand then Gx∈(K). By the Tube Theorem 2.1, there is a G-
invariant open neighbourhood U⊂Mof x, locally modelled by a fixed associated
bundle G×GxN0, in which xreads [(e, 0)].
By definition of the adherence, there is a sequence (xn)n∈N⊂Mβconverging
to xin M, with stabilizers Gxn∈(H). For nbig enough, xn∈Uand it can thus
be identified with some point [(gn, νn)] ∈G×GxN0. The stabilizer of [(gn, νn)] is
G[(gn,νn)] =gn(Gx)νng−1
n
and is thus conjugate to a proper subgroup of Gx, because by assumption Mα
and Mβare disjoint. Since Gx∈(K)and Gxn∈(H), it follows that (K)≺conj
(H).
Stratification of a convex polytope by open faces
There is a natural stratification of a convex polytope into vertices, edges and
higher dimensional faces. Let ∆⊂(Rn)∗be a n-dimensional convex polytope.
Let X1, . . . , Xdin Rnbe the outward-pointing normal vectors to the facets. Then
there exists real numbers λ1, . . . , λdsuch that ∆reads
∆ =
d
\
i=1
{µ∈(Rn)∗| hµ, Xii ≤ λi}.
Let Bbe the set of subsets (possibly empty) β⊂ {1, . . . , d}.For each β∈ B
we consider the intersection
Fβ=\
i∈β
{µ∈∆| hµ, Xii=λi}.
13
If Fβ6=∅, its relative interior ◦
Fβis called a l-dimensional open face of ∆where l
is equal to nminus the cardinality of β. We equip Bwith the strict partial order
α≺β⇐⇒ α6=βand ◦
Fα∩◦
Fβ6=∅.(6)
With this strict partial order, the collection {◦
Fβ|β∈ B} forms a B-stratification
of ∆. A strict partial order is defined on the set of faces by
◦
Fα<◦
Fβ⇐⇒ α≺β.
Finally note that, if α≺βthen β⊂α.
4. G-tubular covers
If (M, G)is a proper G-manifold, the Tube Theorem 2.1 allows us to produce G-
categorical open subsets in the following way: any m∈Madmits a G-invariant
neighbourhood U⊂Msuch that the map ϕ:Y0→Udefined in (2) is a G-
equivariant diffeomorphism. Here
Y0=G×GmN0
where N0is a fixed neighbourhood of zero in some subspace N⊂TmM, com-
plementary to g·min TmM, on which Gmacts linearly. The proper G-manifold
Y0is a local model for U, in which mreads ϕ−1(m) = [e, 0]. The G-homotopy
F:Y0×[0,1] →Y0defined by
F([(g, ν )], t) = [(g, (1 −t)ν)].
is a G-deformation retract of Y0onto the orbit G·[e, 0]. By using the fact that
ϕis a G-equivariant diffeomorphism, the open subset U=ϕ(Y0)is G-categorical
since the G-homotopy H:U×[0,1] →Ugiven by
H(p, t) = ϕF(ϕ−1(p), t)(7)
is a G-deformation retract of Uonto G·m.
Definition 4.1.AG-categorical open subset U⊂Mas above, with associated
G-deformation retract as in (7), is called a G-tubular open subset of M. A cover
of Mmade of G-tubular open subsets is called a G-tubular cover of M.
14
Minimal G-tubular covers
Clearly, every m∈Madmits a neighbourhood which is a G-tubular open subset
of M. Consequently, G-tubular covers of Malways exist. The question is whether
they can be refined. Let Ube any G-tubular cover of M. We know that Mcan be
decomposed into the disjoint union of its orbit-type strata {Mβ|β∈ B}, which
form themselves a smooth stratification of M. Let B0⊂ B be the biggest subset of
indices β∈ B such that Mβis minimal with respect to (4). Consider the disjoint
union M0of all the strata Mβwith β∈ B0. From Uwe extract a subcover U0,
chosen as small as possible such that U0covers M0. In particular U0is a refinement
of U. We ask the following:
(Q) Does it exist a subset U0⊂ U, obtained as above, which covers M?
The answer is in general negative (cf. Example 4.3). However it is positive
for all the proper G-manifolds listed in Example 2.3, where U0is constructed
explicitly.
Definition 4.2.Let (M, G)be a proper G-manifold. U0obtained as above is called
aminimal G-tubular cover if the following holds:
(i) U0is a cover of M.
(ii) For each minimal orbit-type stratum Mβ, the set
V0
β={Vβ=U∩Mβ|U∈ U0}
is the smallest cover by G-categorical open subsets of Mβ, where the topology
of Mβis the subset topology.
We discuss the simplest example where such a cover exists. Let S2⊂R3,
on which S1acts by rotations about the z-axis. This action has two minimal
orbit-type strata, namely the North and South pole. Two small disks centered
at those points are S1-tubular open subsets and can be taken sufficiently big so
that they form a minimal S1-tubular cover of S2. In this example, a disk centered
at the North pole can be extended until its closure meets the South pole. The
impossibility to extend it further relies on the fact that such neighbourhoods
are constructed by mean of the Riemannian exponential map. This map is no
15
longer injective if the disk contains two opposite points on the sphere. The next
proposition gives another answer to this fact by using the properties of G-tubular
open subsets.
Proposition 4.1.Let (M, G)be a proper G-manifold. If U⊂Mis a G-
tubular open subset which intersects a minimal orbit-type stratum Mβ, then
Uretracts onto the orbit G·xof some x∈Mβ. In particular G-tubular open
subsets intersect at most one minimal orbit-type stratum.
Proof — Let β= ((H),(b)) ∈ B such that Mβis a minimal orbit-type stratum.
Let U⊂Mbe a G-tubular open subset of Msuch that U∩Mβ6=∅, and let
H:U×[0,1] →Ube a G-deformation retract of Uonto G·xfor some x∈M.
By contradiction, assume that x∈Mαfor some α= ((Gx),(a)) 6=β.
Each point y∈U∩Mβhas stabilizer Gy∈(H). By G-equivariance of the
homotopy, Gyis a subgroup of GH1(y)which is itself conjugate to Gx, as H1(y)
and xlie on the same orbit. In particular (Gx)≺conj (H). Two cases occur:
(i) If Mβ∩Mα6=∅, then β≺αsince β6=α. By Lemma 3.2 we get (H)≺conj
(Gx)which is a contradiction.
(ii) If Mβ∩Mα=∅we must use the assumption that Uis G-tubular. Let G×Gx
N0be the local model for U. Given y∈Mβwe define the G-equivariant
path y(t) = Ht(y), where t∈[0,1]. In the local model, yreads [g, ν]and y(t)
reads [g, νt]where νt= (1 −t)ν. We can assume without lost of generality
that (Gx)ν=H. Observe that, by linearity of the Gx-action on N0, we have
(Gx)νt= (Gx)ν=Hfor all t6= 1. Hence G[g,νt]=g(Gx)νtg−1=gHg−1
for every t6= 1.In particular, y(t)∈Mβfor all t6= 1. Since the path y(t)
starts at y∈Mβand ends on G·x⊂Mα, there is some t0∈[0,1] such that
y(t0)∈Mα. The parameter t0is chosen the smallest such that this occurs.
If t06= 1, the previous argument shows that y(t0)∈Mβ∩Mα, which is a
contradiction. Otherwise, since y(t)∈Mβfor all t<t0, there is a sequence
(yn)n∈N⊂Mβwhich converges to y(t0). By closedness of Mβ, this yields
y(t0)∈Mβ∩Mα, which is again a contradiction. We conclude that x∈Mβ.
16
The answer to question (Q) is in general negative. In the example below,
(M, G)is a proper G-manifold with Mcompact, and the action admits only one
minimal orbit-type stratum Mβwhich is a single G-orbit.
Example 4.3.Think of M=S3as the set of unit vectors (z1, z2)∈C2equipped
with the S1-action
θ·(z1, z2)=(eiθ z1, e2iθ z2).
This action has only one minimal orbit-type stratum Mβwith β= ((Z2),(b)) for
some index b∈ BZ2. Explicitly
Mβ=n(0, z2)∈C2| |z2|2= 1o
which is diffeomorphic to a circle. In particular Mβis the S1-orbit of the point
(0,1) ∈S3. The S1-invariant open subset
U=(z1, z2)∈S3| |z1|2<2
3
is an S1-invariant tubular neighbourhood of the minimal orbit-type stratum, and
is diffeomorphic to a solid torus (cf. Figure 6). Since Mβis an S1-orbit, Uis an
S1-tubular open subset. We may choose U0={U}. This cover satisfies (ii) of
Definition 4.2 but it does not satisfy (i), since it does not cover M. To cover M
we require the additional open subset
V=(z1, z2)∈S3| |z1|2>1
3
which is also a solid torus, understood as an S1-invariant tubular neighbourhood
of the S1-orbit of (1,0) (cf. Figure 6). It is therefore S1-categorical and then
CatS1(M)≤2. There is in fact equality because otherwise it would mean that
S3is contractible onto a circle, which is untrue.
5. Localization Formula
In this section we obtain a localization formula (cf. Corollary 5.2) for proper G-
manifolds which admit a minimal G-tubular cover. This formula says in particular
17
Figure 6: Representation in R3of the sphere S3with
a point removed. The stratum Mβis a circle closing
at infinity and the tori around it form a solid torus,
which is a tubular neighbourhood.
that the equivariant LS-category of a proper G-manifold is intrinsic to the equiv-
ariant LS-category of its minimal orbit-type strata. The theorem below holds in
general, without any assumption on the proper G-manifold.
Theorem 5.1.Let (M, G)be a proper G-manifold and write Mas the dis-
joint union of its orbit-type strata {Mβ|β∈ B}. Let B0be the biggest subset
of Bsuch that Mβis minimal for every β∈ B0. Then
CatG(M)≥X
β∈B0
CatG(Mβ).
Proof — Let Ube a G-tubular cover of M. Choose U∈ U such that U∩Mβ6=∅
for some β∈ B0, say β= ((H),(b)). By Proposition 4.1,Udoes not intersect any
other minimal stratum and the G-deformation retract H:U×[0,1] →Uretracts
onto an orbit G·xof some x∈Mβ. The set Vβ=U∩Mβis open in Mβfor the
subset topology, and it is G-invariant because so are Uand Mβ.
Let G×GxN0be the local model for U. Given y∈Vβwe define the G-
equivariant path y(t) = Ht(y), where t∈[0,1]. In the local model, yreads [g, ν ],
18
and y(t)reads [g, νt]where νt= (1 −t)ν. Since (Gx)ν∈(H), we use the linearity
of the Gx-action on N0, to obtain (Gx)νt= (Gx)ν= (H)for all t∈[0,1]. Hence
G[g,νt]=g(Gx)νtg−1∈(H)for all t∈[0,1].
In particular, y(t)∈Mβfor all t∈[0,1]. Because y∈Vβis arbitrary and [0,1] is
compact, the map F:Vβ×[0,1] →Vβgiven by Ft(y) = y(t)is a homotopy. It is
clearly G-equivariant by construction and defines a G-deformation retract of Vβ
onto G·x. It follows that Vβis G-categorical.
Let Uβ⊂ U be the subset of all U∈ U such that U∩Mβ6=∅. Then
Vβ={Vβ=U∩Mβ|U∈ Uβ}
is a cover of Mβby G-categorical open subsets, which is not necessarily a minimal
cover. This procedure associates to each β∈ B0a cover Vβof Mβ.
Proposition 4.1 says that, if α, β ∈ B0are distinct, then Uα∩ Uβ=∅. In
particular, each Vβ∈ Vβis determined by a unique U∈ Uβ. Therefore
CatG(M)≥X
β∈B0
CatG(Mβ).
Theorem 5.1 had already been obtained by Hurder and Töben (cf. [13]
Theorem 3.7). However if (M, G)admits a minimal G-tubular cover, the following
occurs:
Corollary 5.2 (Localization Formula).Let (M, G)be a proper G-manifold
which admits a minimal G-tubular cover. Decompose Minto its orbit-type
strata {Mβ|β∈ B}. Let B0be the biggest subset of Bsuch that Mβis
minimal for every β∈ B0. Then
CatG(M) = X
β∈B0
CatG(Mβ).
Proof — By Theorem 5.1, CatG(M)≥Pβ∈B0CatG(Mβ).The other inequality is
a direct consequence of the properties of a minimal G-tubular cover (cf. Definition
4.2).
19
Proposition 5.3.Let (M, G)be a proper G-manifold which admits a min-
imal G-tubular cover. Assume Mβis a minimal orbit-type stratum with
β= ((H),(b)). Then
CatG(Mβ) = CatNG(H)(MH,b).
Proof — Let U0be a minimal G-tubular cover of Mand let Mβbe a minimal
orbit-type stratum. By definition of U0, the set Vβ={Vβ=U∩Mβ|U∈ U0}
is the smallest cover by G-categorical open subsets of Mβ, where the topology of
Mβis the subset topology.
For every Vβ∈ Vβ, let e
Vβ=Vβ∩MH,b. Then e
Vβis an NG(H)-invariant
open subset of MH,b , for the subset topology. Let H:Vβ×[0,1] →Vβbe a
G-deformation retract of Vβonto some orbit G·xof x∈Mβ. Then the NG(H)-
homotopy F:e
Vβ×[0,1] →e
Vβdefined by
Ft=Hte
Vβ
for each t∈[0,1]
is an NG(H)-deformation retract of e
Vβonto the orbit NG(H)·x. Therefore the set
e
Vβ={e
Vβ=Vβ∩MH,b |Vβ∈ Vβ}is a cover of MH,b made of NG(H)-categorical
open subsets. This cover is minimal by assumption and because Mβ=G·MH,b.
We thus get
CatG(Mβ) = CatNG(H)(MH,b).
The reader is invited to compare the above result with [17] (Proposition 2.1).
Example 5.1.We verify Theorem 5.2 on the examples discussed above.
(i) Let M=S2⊂R3on which S1acts by rotations about the z-axis. The
minimal strata have orbit-type (H)where H=S1, namely
Mβb=n(0,0,(−1)b−1)o, βb= ((S1),(b)) and b= 1,2.
Then
CatS1(Mβ1) + CatS1(Mβ2) = 1 + 1 = CatS1(M).
20
(ii) Let M=CP2equipped with the action of S1
θ·[z0:z1:z2] = [eiθz0:z1:z2].
The minimal strata Mβ1and Mβ2have orbit-type (H) = (S1). There is a
CP1and the single point [1 : 0 : 0], respectively. Therefore
CatS1(Mβ1) + CatS1(Mβ2) = 2 + 1 = CatS1(M).
(iii) Let M=S2acted on by the group Tas in Example 2.3 (iii). There are
three minimal orbit-type strata. Two of them, Mβ1and Mβ2, have orbit-type
(Z3). The last minimal stratum Mαhas orbit-type (Z2). We find
CatZ3(Mβ1) + CatZ3(Mβ2) + CatZ2(Mα) = 1 + 1 + 1 = CatT(M)
6. Tubular covers of symplectic toric manifolds
In this section we show that symplectic toric manifolds admit a minimal tubular
cover. Such a cover is constructed explicitly in Theorem 6.1. Let T be torus with
Lie algebra tand dual Lie algebra t∗. The smooth action of T on a symplectic
manifold (M, ω)is Hamiltonian if there exists a momentum map ΦT:M→t∗.
The quadruple (M, ω, T,ΦT)is called a Hamiltonian T-manifold.
Definition 6.1.Let T be an n-dimensional torus. A Hamiltonian T-manifold
(M, ω, T,ΦT)is called a symplectic toric manifold if (M, ω)is a 2n-dimensional
compact connected symplectic manifold and the Hamiltonian action of T on M
is effective.
For symplectic toric manifolds, the image ΦT(M)of the momentum map is
aDelzant polytope i.e. a convex polytope ∆⊂(Rn)∗which is simple i.e. each
vertex xmeets exactly nedges, rational i.e. the edges meeting at a vertex xare
of the form x+tαx,i where αx,i ∈(Zn)∗,smooth i.e. for each vertex xthe isotropy
weights αx,1, . . . , αx,n form a Z-basis of (Zn)∗.
This observation is due to Delzant (cf. [7] Lemmas 2.2and 2.4). Delzant
also proved that ∆determines entirely the symplectic toric manifold (M, ω, T,ΦT),
up to T-equivariant symplectomorphisms (cf. [7] Theorem 2.1). His proof relies
on a well-known result of convexity obtained independently by Atiyah [1] and
21
Guillemin and Sternberg [12], which states that the image of a momentum
map for the action of a torus (not necessarily effective) on a compact connected
symplectic manifold is a convex polytope.
We recall some standard facts about Morse theory applied to a symplectic
toric manifold (M, ω, T,ΦT). The reader is referred to the book of Guillemin
and Sjamaar (cf. [11] Section 3.6) for details. Let MTbe the fixed point set of
T. For every m∈MT, the torus acts on the tangent space at m. There is a T-
invariant complex structure on Msuch that TmMis a complex T-representation
with weight space decomposition
Cαm,1⊕ · · · ⊕ Cαm,n
where αm,1, . . . , αm,n ∈t∗are the weights of the representation. A generic com-
ponent of the momentum map ΦT:M→t∗is a component φξ=hΦT(·), ξiwhere
ξ∈gis generic i.e. αm,i(ξ)6= 0 for every m∈MTand i= 1, . . . , n. In this case,
the critical points of φξare isolated and φξis a Morse function whose critical set
is precisely MT. Moreover every critical point of φξhas even index. Therefore
symplectic toric manifolds possess an extra structure given by the properties of
the T-action. This structure is used to construct a minimal T-tubular cover of
M.
Theorem 6.1.Let (M, ω, T , ΦT)be a symplectic toric manifold. Then M
admits a minimal T-tubular cover.
Proof — Let {Mβ|β∈ B} be the B-stratification of Minto orbit-type strata,
with strict partial order (5). Since T is compact, there are only finitely many
minimal orbit-type strata Mβ1, . . . , Mβ`. By assumption on the T-action, each Mβi
is an isolated fixed point mi∈MT. Then there is ξi∈tsuch that φξiis a generic
component of the momentum map, which takes its minimum at mi. Let −∇φξi
be the gradient vector field associated to this component, with corresponding flow
ϕt. Since the image of the momentum map ΦT(M)is a Delzant polytope ∆, the
B0-stratification {◦
Fβ0|β0∈ B0}of ∆by open faces (cf. Section 3) coincides with
the B-stratification by orbit-type of M. In other words, for every i= 1, . . . , `, we
can associate to βi∈ B a unique index β0
i∈ B0such that ΦT(Mβi)is precisely the
22
zero-dimensional face ◦
Fβ0
i. For each other index α∈ B there is a unique α0∈ B0
such that ΦT(Mα) = ◦
Fα0. Define an open subset Vβ0
i⊂t∗by
Vβ0
i=[
β0
iα0
◦
Fα0.
By continuity and T-invariance of ΦT, the subset Uβi= Φ−1
T(Vβ0
i)is a T-invariant
open neighbourhood of miin M. It reads
Uβi=[
βiα
Mα.
For every m∈Uβi\ {mi}, the flow line ϕt(m)is defined for every t∈R, by
compacity of M. By construction of Uβi, the point mbelongs to some orbit-type
stratum Mαwith βi≺α. Since ϕtis stratum-preserving, ϕt(m)∈Uβifor every
t∈R. Moreover the only critical point of φξiin Uβiis mi, and it is a minimum.
Hence ϕt(m)tends to mias ttends to infinity. Therefore the continuous map
fm: [0,1[ −→ Uβi
t7−→ ϕt
1−t(m)
extends by continuity into a map e
fm: [0,1] →Uβiwith e
fm(1) = mi. Then the
map
H:Uβi×[0,1] −→ Uβi
(m, t)7−→ e
fm(t)
is a T-deformation retract of Uβionto the orbit T ·mi=mi. In particular Uβiis
T-categorical for every i= 1, . . . , `. It is clear that U={Uβi}`
i=1 is a cover of M
made of T-tubular open subsets, which are themselves tubular neighbourhoods of
the closed strata. By Proposition 4.1, this cover is the smallest that we can take
and then Uis minimal.
The result below, due to Bayeh and Sarkar (cf. [4] Theorem 5.1), is then
a direct consequence of the Localization Formula.
23
Corollary 6.2 ([4] Theorem 5.1).Let (M, ω, T , ΦT)be a symplectic toric
manifold. Then CatT(M)coincides with the cardinality of MT.
Our choice to consider symplectic toric manifolds makes the proof of Theorem
6.1 relatively straightforward for two reasons. The first reason is that the fixed
points of the T-action are isolated, and the second reason is that the stratification
by orbit-type strata of Mcoincides with the stratification by open faces of the
polytope.
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School of Mathematics
University of Manchester
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