On relations between gradient and classical equivariant homotopy groups of spheres

We investigate relations between stable equivariant homotopy groups of spheres in classical and gradient categories. To this end, the auxiliary category of orthogonal equivariant maps, a natural enlargement of the category of gradient maps, is used. Our result allows for describing stable equivariant homotopy groups of spheres in the category of orthogonal maps in terms of classical stable equivariant groups of spheres with shifted stems. We conjecture that stable equivariant homotopy groups of spheres for orthogonal maps and for gradient maps are isomorphic.


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K. Gȩba and M. Izydorek JFPTA degree for S 1 -equivariant gradient maps and used it to obtain a global bifurcation theorem applicable to cases where classical results give only limited information. This abstract result has been next applied to obtain a bifurcation theorem for periodic solutions of Hamiltonian systems, for a problem in elasticity and for elliptic equations on domains that admit S 1 -symmetry. In fact our work is motivated by [6]. The aim of this paper is better understanding the homotopy-theoretical background of topological invariants associated with equivariant and gradient equivariant maps. More precisely, we investigate relations between gradient and nongradient equivariant homotopy groups of spheres. When trying to understand these relations it is convenient to consider yet another class-the class of orthogonal equivariant maps. That class is in some sense natural enlargement of the class of gradient equivariant maps. We are concerned with decomposition results in categories of equivariant maps, orthogonal equivariant maps and gradient equivariant maps. Our result, Theorem 1.3, allows for describing stable equivariant homotopy groups of spheres in the category of orthogonal maps in terms of classical stable equivariant groups of spheres with shifted stems. We conjecture that stable equivariant homotopy groups of spheres for orthogonal maps and for gradient maps are isomorphic. We would like to point out that methods developed in this paper allow for a simultaneous proof of the decomposition result (Theorem 1.1) in all three categories of mappings under consideration. The concept of otopy, introduced by Becker and Gottlieb in [3], provides a convenient framework for our proofs.
In what follows, every subgroup H of a compact Lie group G is understood to be closed and (H) stands for a conjugacy class of H. If NH is the normalizer of H in G, then the quotient group W H = NH/H is called the Weyl group (see [13]). Throughout the paper, R k denotes the trivial representation of G and I is the unit interval [0, 1] with the trivial action of G. Finally, we set We will be concerned with the following three families of maps. ( G is a set of corresponding homotopy classes. and a G-map ϕ : U → R such that [S k+V , S V ] ∇ G is a set of homotopy classes of equivariant gradient maps For an arbitrary G-representation X, a linear subspace X H of X consisting of points fixed by H is a representation of the Weyl group W H.
The isotropy type of an invariant subset Ω of a representation X is the set Here we follow the terminology of [4]. Accordingly, Iso(Ω) is partially ordered, (H) ≤ (K), if and only if H is conjugate to a subgroup of K. The free isotropy type will be denoted by (e) instead of ({e}). In particular, X e denotes a subset of X on which G acts freely. Assume that (e) ∈ Iso(V ). Put A k+V = S k+V \ S k+V e . We will also consider the maps f ∈ M • G (S k+V , S V ) that send A k+V to a basepoint. The set of corresponding relative homotopy classes is denoted by Our first result states as follows.

Theorem 1.1. There is a natural bijection
In the case k + dim V G ≥ 2, this is an isomorphism of abelian groups: where • means * , ⊥ or ∇.
The case • = * is well known. One can find corresponding results in [8], [10], [11] and [12] and even more general theorems in [9]. However, our approach is different and works in each of the above three cases with analogous proofs.
Let [S X ; S Y ] • G be a set of homotopy classes of maps in one of the categories • = * , ⊥ or ∇.
The smash product with the identity map · ∧ (id : The following theorem is an immediate consequence of Theorem 1.1.

Theorem 1.2. There is a natural isomorphism of abelian groups
where • means * , ⊥ or ∇. JFPTA Next, we are going to explain relations between sets of homotopy classes Here is our result.
There is a natural bijection (1.4) In the case k + dim V G ≥ 2, this is an isomorphism of abelian groups.
The following result on stable equivariant homotopy groups is a consequence of Theorem 1.3.

From homotopy to otopy
In the proof of Theorem 1.1 certain properties of the so-called equivariant local maps will be crucial (see [2]). We

Local maps
for each (x, v) ∈ U . The set of all local G-equivariant orthogonal maps on Ω is denoted by F ⊥ G (Ω), and F ⊥ G [Ω] stands for the corresponding set of otopy classes.
A local G-equivariant gradient map is a pair (f, U ) ∈ F * G (Ω) such that there are a G-invariant open neighbourhood U of f −1 (0), U ⊂ U , and a G-invariant map ϕ : U → R such that Vol. 12 (2012) Gradient and classical equivariant homotopy 53 F ∇ G (Ω) denotes the set of all local G-equivariant gradient maps on Ω, and F ∇ G [Ω] is the corresponding set of otopy classes. Note that for every local G-equivariant map (f, U ) ∈ F • G (Ω), where • = * , ⊥ or ∇, there is an otopic local G-equivariant map with relatively compact domain. It is enough to take appropriate restriction of f . Furthermore, (f, U ) can be modified within its otopy class to a local G-equivariant map (f, U) that is proper. This result was proved in [2] (see Theorem 7.1) in the case • = * . Cases • =⊥ and • = ∇ can be proved analogously. With is continuous. Using this observation we will identify G-equivariant maps The bijection R * 1 is proved in [2] (see Theorem 7.1). The other cases can be proved analogously.

The splitting lemma
We say that an open invariant subset U ⊂ Ω is (H)-normal if there exists ε > 0 such that ν defines a diffeomorphism U ε (U (H) ) → U . Note that if U is (H)-normal, then every element of it has a unique representation

Proof of Theorem 1.2
Let V ⊂ W = V ⊕ R n be representations of a compact Lie group G.
Notice that Iso(V ) = Iso(W ). (2. 3) The following remark is a direct consequence of (2.3) and the definition of A Ω,H .