Contemporary Mathematics
Equivariant LS-category for finite group actions Hellen Colman
1. Introduction In this paper we study the equivariant category of finite group actions. We introduce the basic filtration for the orbit space of the action. In terms of this filtration we give upper and lower estimates of the equivariant category. The idea for the proof is parallel to the approach in [3] for compact-Hausdorff foliations. We give examples to show that both the upper and lower bounds are realized. It is interesting to compare these estimates with the upper bound given by Marzantowicz in [11]. We present an example that contradicts that estimate. The author is grateful to John Greenlees for many helpful conversations. The ideas of this paper were developed during the author’s stay at the University of Sheffield in Fall 2000, and she would like to thank the members of Department of Mathematics for their warm hospitality. 2. Equivariant category Let G be a compact Lie group acting on a topological space X. An invariant subspace U of X is G-categorical if there exists an equivariant homotopy H : U ×I → X such that H0 is the inclusion and H1 : U → X has image in a single orbit.
Definition 2.1. The equivariant category, catG X, is the least number of Gcategorical invariant open sets required to cover X. If no such covering exists we say that the equivariant category is infinite.
The equivariant category is an invariant of equivariant homotopy type that coincides with the classical category when the action of G is trivial on X. If the action is free, we have catG X = catX/G. In general, the category of the orbit space is just a lower bound. Proposition 2.2. [6, 11] catG X ≥ catX/G.
For an orbit G/H its orbit type is the conjugacy class (H) of H. If the set of orbit types is finite, we say X has finite orbit type. For spaces of finite orbit type, Marzantowicz gives an upper bound for the equivariant category in terms of the dimension of X/G and the number of connected Partially supported by grant EU RTN1-1999-00176, University of Sheffield, Modern Homotopy Theory. c !0000 (copyright holder)
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components of the minimal set. The Marzantowicz minimal set is the subset of X/G corresponding to the orbits of type (H) such that there is no point x ∈ X with isotropy group Gx ⊇ \ H.
Marzantowicz’s Theorem 2.3. [11] If X is a G-space of finite orbit type and α is the number of connected components of the minimal set, then catG X ≤ α(dim X/G + 1). We will present a counterexample to this Theorem in Section 4. The purpose of this paper is to provide a new upper bound for catG X in terms of the dimension of X/G and the G-category of the singular set of the action. Useful lower bounds for catG X can be obtained from estimates derived from cohomology cup product lengths. Following the approach of Marzantowicz and Ramsay [11, 12], consider the Borel cohomology HG (X) = H(EG ×G X) where EG is a contractible space on which G acts freely. If all the orbits have trivial Borel cohomology, then ˜ ∗ X. catG X ≥ nilH G The assumption that all the orbits satisfy the dimension axiom is very restrictive, and to avoid this there are a variety of different approaches in the literature which partially overlap [4, 5, 6]. Finally the G-category of a manifold M can be used to estimate the number of critical orbits of a G-invariant differentiable function f : M → R. Theorem 2.4. [6] Let M be a compact G-manifold and f : M → R a Ginvariant differentiable function. Then catG M ≤ number of critical orbits of f. 3. The basic filtration From now on, we assume G is a finite group. If H is a subgroup of G, we write H ≤ G. Let H be the set of all subgroups of G. We introduce a finite filtration of H: H = H0 ⊃ H1 ⊃ · · · ⊃ Hr = {G} where Hi+1 = {H ≤ G| ∃Hi ∈ Hi such that H ⊇ \ Hi } for each 0 ≤ i ≤ r − 1. A group H ∈ Hi exactly if there is a subgroup chain of length l ≥ i such that {id} ⊆ / H2 · · · ⊆ / Hl = H. / H1 ⊆ Let B = X/G be the orbit space of the action of G on X, where G is a finite group. Its singular set is the image of the points with non trivial isotropy in X. The fixed point set of a G-space with respect to a subgroup H is denoted by X H and its image in the orbit space by B H . We will construct a filtration of the space B. For each 0 ≤ i ≤ r, let ! Xi = X H and B i = X i /G. H∈Hi
We have that B is the singular set and B r = B G the image of the fixed set of the action (possibly empty). Let I = {i| B i is finite} and J = {j| B j+1 = ∅}. We define k ∈ Z as min I, if I ,= ∅; min J, if I = ∅ and J ,= ∅; k= r, otherwise. 1
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Finally, we introduce the basic filtration of length k of B: B = B0 ⊃ B1 ⊃ · · · ⊃ Bk
where B k is either discrete or the Marzantowicz minimal set. 4. Estimates From now on, we assume the action is not free (for free actions the G-category coincides with the category of the orbit space). We show that the G-category of X is bounded below by the ordinary category of any term of the filtration of B. The proof is based on the fact that any equivariant homotopy must respect the filtration of X. Lemma 4.1. Let U be an invariant open subset of the G-space X and H : U × I → X an equivariant homotopy. Then for i = 1, · · · , r we have Ht (X i ∩ U ) ⊂ X i for all t ∈ I. Proof. Let x ∈ X i ∩ U and x# = Ht (x). We have x ∈ X H for some H ∈ Hi since x ∈ X i , so gx = x for all g ∈ H. Then x# = Ht (x) = Ht (gx) = gHt (x) = gx# for all g ∈ H. So x# ∈ X i since # x ∈ X H with H ∈ Hi . ! Proposition 4.2. catG X ≥ catG X i for all i = 1, · · · , r.
Proof. Suppose U is G-categorical for X and H : U × I → X is an equivariant homotopy of the inclusion. Let V = U ∩ X i be the relatively open subset (possibly empty) of X i . By Lemma 4.1, we can consider the equivariant homotopy H|U ∩X i : (U ∩ X i ) × I → X i and V is G-categorical for X i . ! Corollary 4.3. catG X ≥ catB i for all i = 1, · · · , r.
Proof. Since the orbit space of the action restricted to each X i is X i /G = B i , we have catG X i ≥ catB i . !
In particular, catG X ≥ catB k where B k is the first discrete term of the filtration of B or, in case that there are no discrete terms, the Marzantowicz minimal set. Whenever the first discrete term of the filtration does not coincide with the minimal set, our lower bound gives a counterexample to the upper bound given in Marzantowicz’s Theorem 2.3. Example 4.4. Z4 -action on a surface of genus 5. Consider X a surface with genus 5 as a connected sum of a torus and two surfaces of genus 2. Rotate one of the surfaces 90 degrees before connecting, as in Figure 1.
Figure 1
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I will construct a diffeomorphism τ : X → X with two fixed points. Let τT be a rotation 90 degrees defined on T , the torus with two holes.
Figure 2 This rotation has two fixed points a and b. Let τΣ be a rotation 180 degrees about the z−axis plus a rotation 90 degrees about the x−axis defined on Σ = LΣ ∪ RΣ.
Figure 3 This diffeomorphism has no fixed points. Define τ : X → X such that % τT on T τ= τΣ on Σ
For all x ∈ T ∩ Σ, τT (x) = τΣ (x). Then τ is well defined and has exactly two fixed points which occur in the central torus. We observe that τ 2 is a rotation 180 degrees about the x−axis
Figure 4 and so, it has 12 fixed points. Let G = Z4 = 0τ 1 act on X in the way explained above.
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There are three orbit types: {id}, Z2 and Z4 . The orbit space of the action is the orbifold B = S4422222.
Figure 5 The basic filtration of B has length k = 1 with B 0 = B and B 1 the singular set of the orbifold. The category of B is 2 since it is topologically a sphere and catB 1 = 7, the number of singular points of the orbifold. By Corollary 4.3, catG X ≥ 7. In this case we can choose 7 open sets covering the singular set and extend them to cover the whole space, so catG X = 7. The Marzantowicz upper bound in this case gives catG X ≤ 2(dim X + 1) = 6 since the minimal set is X G = {a, b} and the number of connected components of B G is α = 2. Now, we will give a new upper bound for the G-category in terms of the dimension of B = X/G and the G-category of the singular set. Theorem 4.5. Let X be a connected G-complex, where G is a finite group acting non-freely on X. Then catG X ≤ dim B + catG X 1
Proof. Let {V1 , · · · , Vm } be a G-categorical covering of X 1 . For each invariant open subset Vi of X 1 , there exists a G-categorical invariant open subset Ui of X such that Vi ⊂ Ui (see [6]). So, X 1 ⊂ X is covered by m G-categorical subsets of X. ˆ = B \ B1 The action of G restricted to X \ X 1 is free and its orbit space B is the regular set of B (i.e. the image of the orbits with trivial isotropy). We will ˆ is deformable into a (n − 1)−dimensional simplicial complex and then, show that B ˆ ≤ n. catB Take a small triangulation of B such that the intersection of B 1 with each top dimensional simplex is either contractible or empty [1, 7]. For each top dimensional simplex with empty intersection choose a point p in its interior, let P be the set of these points. Choose a path γp in B for each p ∈ P , which never crosses itself, joining this point p with a point in B 1 such that γp (I) ∩ γq (I) ⊂ B 1 for all q ∈ P , q ,= p. Let ! γp (I) Γ= p∈P
ˆ has the same type of homotopy than B ˆ \ Γ. So, B ˆ The open subset B \ Γ is deformable into its (n − 1)−simplicial structure T , so its category is bounded by the category of T . Since X is a connected G-complex, B is connected and paracompact. Therefore, by the classical dimensional bound ˆ \ Γ ≤ n. for the category [8], we have catB
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ˆ ≤ n and since B ˆ is the orbit space of the free action on X \ X 1 we Then catB ˆ ≤ n. We can choose {Vm+1 , · · · , Vm+n } a G-categorical have catG X \ X 1 = catB covering of X \ X 1 and {V1 , · · · , Vm , Vm+1 , · · · , Vm+n } is a G-categorical covering of X. ! Corollary 4.6. Let X be a connected G-complex, where G is a group of order p prime acting non-freely on X. Then, max{catB, catB G } ≤ catG X ≤ n + catB G .
The last example realizes both the lower and upper bounds in the estimate above. Example 4.7. Projective plane
Consider G = Z3 and X the projective plane RP2 . Define the action φ : Z3 × RP → RP2 to be the quotient of a rotation by 2π/3 on the covering 2-sphere S2 . The rotation has 2 fixed points on S2 , denoted by {±a}. As G has odd order, the quotient action on RP2 has a unique fixed-point, denoted by [a]. Then catB G = 1. The orbit space B is identified with RP2 , hence, catB = 3. Thus, 2
3 = catB ≤ catG X ≤ 2 + catB G = 3. References [1] G.E.Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972. [2] H.Colman; E.Macias-Virg´ os. The transverse Lusternik-Schnirelmann category of a foliated manifold, Topology 40 (2) (2001), 419-430. [3] H.Colman; S.Hurder. LS-category of compact-Hausdorff foliations, preprint, June 2000. [4] M.Clapp; D.Puppe. Invariants of the Lusternik–Schnirelman type and the topology of critical sets, Trans. Amer. Math. Soc. 298 (1986), 603-620. [5] M.Clapp; D.Puppe. Critical point theory with symmetries, J. reine angew. Math. 418 (1991), 1-29. [6] E.Fadell. The equivariant Ljusternik-Schnirelmann method for invariant functionals and relative cohomological index theories, In M´ ethodes Topologiques en Analyse Non-Lineaire, ed. A. Granas, Montreal, 1985. [7] S.Illman. Smooth equivariant triangulations of G−manifolds for G a finite group, Math. Ann. 233 (1978), 199-220. [8] I.M.James. On category, in the sense of Lusternik-Schnirelmann, Topology 17 (1978), 331348. [9] I.M.James. Lusternik-Schnirelmann category, Chapter 27 Handbook of Algebraic Topology, Elsevier Science, Amsterdam, 1995, 1293-1310. [10] L.Lusternik and L.Schnirelmann. M´ ethodes topologiques dans les Probl` emes Variationnels, Hermann, Paris, 1934. [11] W.Marzantowicz. A G-Lusternik-Schnirelmann category of space with an action of a compact Lie group, Topology 28 (1989), 403-412. [12] J.R.Ramsay, Extension of Ljusternik-Schnirelmann category theory to relative and equivariant theories with an application to an equivarian critical point theorem, Topology and its Applications 32 (1989), 49-60.
Hellen Colman Department of Mathematics (m/c 249) University of Illinois at Chicago 851 S. Morgan St. CHICAGO, IL 60607-7045 USA Email:
[email protected] Web: http://www.math.uic.edu/∼hcolman/
