On the space of equivariant local maps

Abstract

We introduce the space of equivariant local maps and present the full proof of the splitting theorem for the set of otopy classes of such maps in the case of a representation of a compact Lie group.

A Hopf type theorem for equivariant local maps

“…The following result is an easy consequence of Propositions 3.2 and 3.9. In [1] we proved the following result. Recall that if dim G > 0 then the set F G [Ω] has a single element (see [1,Theorem 3.1]).…”

“…Any element of Loc G (I ×Ω, V, 0) is called an otopy and any element of Prop G (I ×Ω, V ) is called a proper otopy. In [1] we proved that each otopy (resp. proper otopy) corresponds to a path in F G (Ω) (resp.…”

“…It is well-known that the set Iso(Ω) is finite and so is Iso(Ω) ∩ Φ k (G). The following splitting result was proved in [1] and [4].…”

“…Our paper contains also some remarks concerning the parametrized case. Although the consideration of the parameters may seem to be an artificial generalization, but, in fact, it is a natural way to study the higher homotopy groups of the space F G (Ω) using the formula F G [R k × Ω] ≈ π k (F G (Ω)) established in [1].…”

The Hopf type theorem for equivariant gradient local maps

“…It may be worth noting the relation of our constructions with the additive subgroups U (V ) and A(V ) of the Euler ring U (G) and the Burnside ring A(G) taking into account only their additive structure. Namely, denoting by Θ U the function Θ from Main Theorem, by Θ A its nongradient version from [5], by π U and π A the summing on j and by U the forgetful functor, we obtain the following commutative diagram:…”