Definition of equivariant transversality

The definition of equivariant transversality which was given in [Bie77] relies on some basic results about G-spaces and G-representations, which are essential for the understanding of equivariant maps: the Slice Theorem, Bochner’s lin-earization theorem, and the theory of polynomial generators of invariant and equivariant maps.

By the Slice Theorem, for every x∈M, there is a tubular neighbourhood U 'G×G_x Sx, where Sx is a Gx-invariant slice, such that the set of smooth equivariant maps C_G^∞(M, N) is in one-to-one correspondence to C_G^∞

x(Sx, N):

The inverse of the restriction mapf 7→f _S

x forf ∈C_G^∞(Sx, N) is given by the unique equivariant extensionh7→˜hofg∈C_G^∞

x(Sx, N) with ˜h(gx) =gh(x) for x∈S andg∈G.

Together with Bochner’s linearization theorem (Theorem 1.24), the Slice Theorem (Theorem 1.28) shows that locallyG-equivariant maps can be consid-ered as equivariant maps between representations: By Bochner’s theorem for any x∈m, there is a neighbourhood U of f(x) that is locally G_f(x) -equivari-antly diffeomorphic to theG_f(x)-representation onT_f(x)N. IffisG-equivariant, G_x⊂G_f(x) andT_f(x)N can be considered as aG_x-representation.

Furthermore, we can choose Sx such that f(Sx) ⊂ U. Let Sx be locally Gx-diffeomorphic to the Gx-representationV. Since a small perturbation of f also maps Sx into U, we only need to consider Gx-equivariant smooth maps

fromV intoT_f(x)N to study the stability of the setf⁻¹(P) locally atx. Sup-pose f(x) ∈ P, otherwise we define f to be G-transverse to P at X. Since P is G-invariant, T_f(x)P is a Gx-invariant subspace of T_f(x)N, hence it has a Gx-invariant complementW.

In this way, we reduce the definition to the case of representations and the invariant submanifold 0: We will defineGx-transversality to 0∈W at a point in V^Gx and extend the definition to the general case by calling f G-transverse to P ⊂ N at x ∈ M if the composition of f with the projection to W is G_x-transverse to 0 ∈W. Of course, it has to be shown that this definition is independent of the choice of the slice. We will omit this proof and refer to the literature.

To define transversality to 0 in the case of representations, we need the theory of polynomial generators of invariant and equivariant smooth maps on representations. An introduction to the theory and proofs can be found in [GSS88, chapter XII, §4–6]. (The proof of Schwarz’s theorem is only sketched.

For a complete proof, we refer to Schwarz’s original paper [Schw75].)

Recall that forG-spacesM andN,C^∞(M)^Gdenotes the ring ofG-invariant smooth real-valued functions onM andC_G^∞(M, N) denotes theC^∞(M)^G -mod-ule ofG-equivariant smooth maps fromM toN. Similarly forG-representations V andW, we writeP(V)^Gfor the ring ofG-invariant real-valued polynomials on V andPG(V, W) for theP(V)^G-module ofG-equivariant polynomial maps from V toW, wherepolynomialmeans that the map can be expressed in polynomials in the coordinates with respect to some (and hence any) choice of bases.

First, we need some facts about invariant polynomials and equivariant poly-nomial maps, which can be derived from Hilbert’s basis theorem:

Theorem 6.7 (Hilbert-Weyl Theorem). Let Gbe a compact Lie group and V be aG-representation. Then P(V)^G is finitely generated.

Obviously, the generatorsp1, . . . , pl of PG(V) can be chosen to be homoge-neous. If such a set is minimal (no proper subset is a generating set), it is called aminimal set of homogeneous generatorsofP(V)^G.

Theorem 6.8. PG(V, W)is a finitely generatedP(V)^G-module.

As for the invariant polynomials, it is possible to choose homogeneous gen-erators of PG(V, W): Take the homogeneous parts of any set of generators.

Again, if such a set is minimal, it is a minimal set of homogeneous generators ofP_G(V, W).

Although we do not need it for the definition of equivariant transversality, we start with Schwarz’s theorem, since it will be necessary later on for the higher order version. Moreover, the theorem for equivariant maps may be derived from it.

Theorem 6.9 (Schwarz’s Theorem, [Schw75]). Let G be a compact Lie group andV be a G-representation. If p1, . . . , pl generate P(V)^G, anyf ∈C^∞(M)^G can be written in the form f = h◦P, where P = (p1, . . . , pl) : V → R^l and h∈C^∞(R^l).

The next theorem is due to Malgrange and appeared first in [Poe76]. A possible proof uses Schwarz’s theorem (see [Poe76] or [GSS88]). Another proof is based on an equivariant version of the Stone-Weierstraß theorem (see [Fie07]):

Theorem 6.10. If the equivariant polynomial maps F1, . . . , Fk generate the P(V)^G-module PG(V, W), they also generate the C^∞(V)^G-module C_G^∞(V, W).

Hence, every f ∈C_G^∞(V, W)can be written as f(x) =

k

X

i=1

gi(x)Fi(x), where theg_i are invariant functions.

Example 6.11. Let V = W be the non-trivial Z2-representation RZ2 on R. Then every equivariant map is a product of an invariant function and the func-tionF1(x) =x.

Example 6.12. Let V =RZ2×R, where Z2 acts trivially on the component R, andW =RZ2. Then every equivariant map is also a product of an invariant function and the functionF₁(x, λ) =x.

To define equivariant transversality, we fix a minimal set F1, . . . , Fk of ho-mogeneous generators ofPG(V, W). With respect to this choice, we define the algebraic function

ϑ(x, t1, . . . , tk) = ΣtiFi(x).

Σ :=θ⁻¹(0) is an algebraic set. Any algebraic set admits a canonical Whit-ney stratification. Hence, there is an appropriate definition of transversality to Σ such that the Thom-Mather transversality theorem applies.

For any functionf ∈C_G^∞(V, W), we choose a representation f(x) =

k

X

i=1

g_i(x)F_i(x) and set

Γf(x) = (x, g1(x), . . . , gk(x))∈V ×R^k. Thenf =ϑ◦Γf.

We use this to define G-transversality to 0∈W at 0∈V:

Definition 6.13. f ∈C_G^∞(V, W) is G-transverse to 0 ∈W at 0 ∈V (f tG 0 at 0) iff Γf is transverse to Σ at 0.

Of course, it has to be shown that this is well-defined. In particular, one has to prove the independence of the choice of the Fi and the choice of the representation off as a linear combination of theFi, which is not always unique.

Again, we refer to [Bie77], [Fie96], and [Fie07].

Example 6.14. In Example 6.11, Σ ⊂RZ2 ×R is given by the union of the two lines x= 0 and t = 0. The canonical stratification consists of the point (0,0) and each of the lines with the point (0,0) omitted. Hence f :RZ2 →RZ2

isZ2-transverse to 0 at 0 iff∂_xf(0,0) =g(0,0)6= 0.

Example 6.15. In Example 6.12, Σ⊂(RZ2×R)×Rconsists of the product of the two lines x= 0 andt = 0 with the λ-axis. The canonical stratification is the same but each stratum is multiplied with theλ-axis. In this case, Γf is transverse to Σ at (0,0) if∂xf(0,0) =g(0,0)6= 0 or∂xf(0,0) =g(0,0) = 0 and

∂λ∂xf(0,0) =∂λg(0,0)6= 0. In the second case, the local preimage of the zero set off looks like a pitchfork.

Example 6.15 illustrates the application of equivariant transversality theory to bifurcation theory. We will come back to this in section 6.2.

As explained above, Definition 6.13 is extended to the general case by choos-ing a slice atx∈M and a complement toTf(x)P iff(x)∈P. Independence of choices is proved in [Bie77].

Remark 6.16. For any x ∈V, S = (gx)^⊥ contains a slice for the G-action.

Hence a G-equivariant map f :V → W is G-transverse to 0 ∈ W at x iff its restriction toSisG_x-transverse to 0∈W. Since generators ofP_G(V, W) restrict to generators ofP_Gx(S, W), this is equivalent to Γ_f tΣ atx, see [Bie77].

Based on the theory of transversality to Whitney stratified sets, the transver-sality Theorem 6.2 can be transferred to the equivariant case:

Theorem 6.17 ([Fie07, Proposition 6.14.2 and Theorem 6.14.1]). Let M and N be smoothG-manifolds andP ⊂N be a smoothG-invariant submanifold.

1. Iff :M →N is a smooth G-equivariant map andf t^G P, then f⁻¹(P) is a Whitney stratified subset ofM.

2. If f : M → N is a smooth G-equivariant map and f tG P at x ∈ M, there is a neighbourhoodU ⊂M ofxsuch that f tGP at everyy∈U. 3. If M is compact, the subset T of maps that are G-transverse to P is

residual inC_G^∞(M, N)with respect to theC^∞-topology.

4. IfA⊂M is compact andP is closed, the set

TA:={f ∈C_G^∞(M, N)|f tG P alongA}

isC^∞-open.

5. If M is compact, f : M ×[0,1] → N smooth, ft := f(·, t) t^G P for all t ∈[0,1], then there is an isotopy of G-equivariant homeomorphisms K:M ×[0,1]→M,Kt:=K(·, t), with

Kt(f_t⁻¹(P)) =f₀⁻¹(P) and K0=1M.

Remark 6.18. Again we may omit the compactness assumption onM in 3 and Ain 4 if we replace theC^∞-topology with the WhitneyC^∞-topology. Then we requireAto be closed.

For our application to Hamiltonian relative equilibria, we will need the gen-eralization of the theory to jets of functions, which is also used in bifurcation theory to predict stability properties of the branches: