By Alexander M. Kushkuley, Zalman I. Balanov
The publication introduces conceptually uncomplicated geometric principles according to the lifestyles of basic domain names for metric G- areas. an inventory of the issues mentioned contains Borsuk-Ulam variety theorems for levels of equivariant maps in finite and limitless dimensional situations, extensions of equivariant maps and equivariant homotopy category, genus and G-category, elliptic boundary price challenge, equivalence of p-group representations.
The new effects and geometric rationalization of a number of identified theorems awarded the following will make it fascinating and beneficial for experts in equivariant topology and its functions to non-linear research and illustration theory.
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Additional info for Geometric Methods in Degree Theory for Equivariant Maps
Example text
Then X is a G - E N R - s p a c e iff X H is an E N R - s p a c e for every H E Iso(X). 4. Let a finite group G act on a compact manifold X '~, let A C X ~ be a dosed invariant subset and let the action of G on X ~ \ A be free. Let V be an orthogonal (n + k)-dimensional representation of G (k >_ 1) and B a unit bM1 bounded by the unit sphere S ( V ) = OB. Then any equivariant map f : A ---* S ( V ) has an equivariant extension F : X " --* S ( V ) . 5. Let a finite group G act on a compact smooth manifold X '~, let K C X n be a closed invariant subset and let the action of G on X "~ \ K be free.
Let a finite group G act on a compact smooth manifold X n, let I( C X ~ be a dosed invariant subset and let the action of G on X n \ K be free. Then any neighborhood of I f contains an open invariant neighborhood U such that X \ U is a compact smooth manifold with boundary. Moreover, X \ U possesses a finite invariant triangulation. 5. 3 (here U is an invariant neighborhood of K). Let U C C 28 1. 18. 3 once again extend flu to an equivariant m a p # : U U X "-1 ~ B \ {0} (here X ~-~ is an invariant (n - 1)-skeleton in X ) .
Israilevich and E. 3) for arbitrary p. 2. A "completely homological" proof of the theorem of Krasnoselskii was obtained by Ju. Borisovich and Y. Izrailevich [Boil who used the Borel spectral sequence. 4 was proved (in the 42 2. Topolgicai actions framework of the geometric approach) by P. Zabrejko under the additional "regularity" condition. This condition was dropped by Ju. Borisovich, Y. Izrailevieh and T. Schelokova [BoISe,Sel,Sc2,Sc3,Sc4]. Later on a corresponding geometric proof was suggested by P.