A simple proof of the degree formula for \((\mathbb{Z}/p)\)-equivariant maps
The paper contains elementary and short proofs of several Borsuk-Ulam type theorems. Here are two examples. 1. If \(G= \mathbb{Z}/p\), \(p\) a prime, acts on a normed linear space via linear isometries and if \(f: \overline{\Omega}\subset X\to X\) is a compact perturbation of the identity which is also equivariant then \(\deg(f, \Omega,0)\equiv \deg(f^ G, \Omega^ G,0)\bmod p\). Here \(f^ G\) denotes the restriction of \(f\) to the set of points fixed under \(G\) and it is assumed that these Leray-Schauder degrees are defined. 2. If \(V\) and \(W\) are orthogonal representations of \(G= (\mathbb{Z}/p)^ k\) such that \(V^ G=0= W^ G\) then the existence of an equivariant map \(SV\to SW\) implies \(\dim W\geq \dim V\). Here \(SV\) is the unit sphere of \(V\). There are related results about Lusternik-Schnirelmann category of lens spaces and about the equivariant category of representation spheres. Similar results hold for \(G=(S^ 1)^ k\). Published proofs of these theorems usually apply methods from algebraic topology, in particular if \(G\) is different from \(\mathbb{Z}/2\) or \(S^ 1\). Here only properties of the Leray-Schauder degree and Sard's lemma are being used.
equivariant Lusternik-Schnirelmann category
scientific article; zbMATH DE number 549733