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written down by S.V.Duzhin September 22, 1998

Summary: Problems
written down by S.V.Duzhin
September 22, 1998
Foreword of the publisher
These are the notes of V.I.Arnold's talk at the seminar taken down in vivo. I take
the responsibility for the possible inaccuracies in my interpretation. A more complete
and authentic Russian edition is being prepared by PHASIS.
1. Pseudoring of a Lie group
Let G be a Lie group. Denote by H(G) the set of homotopy classes of continuous
mappings of (G; e) into itself equipped with two operations:
(a  b)(g) = a(g)b(g);
(ab)(g) = a(b(g)):
Although we use additive notation for the rst operation, it may be non-commutative,
e.g. for the group SU(2)  SU(2). The two operations obey one one-sided distributive
law (a + b)c = ac + bc.
Examples. In the following examples, H(G) turns out to be a ring.
1. H(U(1)) = Z.
2. H(O(1)) = Z 2 .
3. H(SU(2)) = Z.


Source: Arnold, Vladimir Igorevich - Steklov Mathematical Institute


Collections: Mathematics