Equivariant degree for Abelian actions. Part I: Equivariant homotopy groups

Brouwer degree, equivariant maps and tensor powers

“…Notice that the statement of Proposition 2.2 does not follow from the well known results for abelian groups (see, for example, [5,6]). Indeed, there are only two abelian subgroups in Q 8 , namely Z 2 and Z 4 for which the "restriction" of the actions leads to the congruencies deg ψ ≡ 4 (mod 2) and deg ψ ≡ 4 (mod 4), respectively.…”

“…It is our intention to elaborate the above construction for certain important representations of classical (infinite) groups (for example, SU (n) and Sp (n)). Observe also that Remarks 2.6 and 2.9 give rise to the question if there is any congruence relation, similar to (1.1), for Hopf invariants of equivariant maps from S 4n−1 to S 2n (see also [3,5,6]). …”

“…We should also mention Example 4.1 from [6] (which was partially inspired by Bartsch's example) describing possible values of Brouwer degrees of equivariant maps "generic" in a certain sense. The problem of existence of decreasing dimension equivariant maps is also related to the "secondary obstructions" in the equivariant extension context (see [5,6,8] for more details).…”

“…We would like to point out that this problem is also closely related to the verification of condition (iii) of Theorem 1.1 (cf. [8] and [5,6]). This paper is intended to reveal common features behind certain examples of equivariant maps related to the above issues.…”

“…We also refer to [5,6] for the discussion and related results in the case where G is an abelian group.…”