An almost classification of compact Lie groups with Borsuk-Ulam properties

Borsuk-ulam Type Theorems for Compact Lie Group Actions

Borsuk-Ulam type theorems for arbitrary compact Lie group actions are proven. The transfer plays a major role in this approach. We present Borsuk-Ulam type theorems for arbitrary compact Lie group actions. The essence of our approach is a generalization of the ideal-valued index of FadellHusseini [FH88] using transfer [Boa66], [BG75], [Dol76], [KP72], [Rou71]. Once an appropriate concept (Defin...

Almost commuting elements in compact Lie groups

The Borsuk-Ulam Theorem

I give a proof of the Borsuk-Ulam Theorem which I claim is a simplified version of the proof given in Bredon [1], using chain complexes explicitly rather than homology. Of course this is a matter of taste, and the mathematical content is identical, but in my opinion this proof highlights precisely where and how the contradiction arises. 1 Statements and Preliminaries Theorem 1.1 (Borsuk-Ulam). ...

The Borsuk-Ulam Theorem

For simplicity, we adopt the following rules: a, b, x, y, z, X, Y , Z denote sets, n denotes a natural number, i denotes an integer, r, r1, r2, r3, s denote real numbers, c, c1, c2 denote complex numbers, and p denotes a point of En T. Let us observe that every element of IQ is irrational. Next we state a number of propositions: (1) If 0 ≤ r and 0 ≤ s and r2 = s2, then r = s. (2) If frac r ≥ fr...

Digital Borsuk-Ulam theorem

The aim of this paper is to compute a simplicial cohomology group of some specific digital images. Then we define ringand algebra structures of a digital cohomology with the cup product. Finally, we prove a special case of the Borsuk-Ulam theorem fordigital images.