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Real algebraic principal abelian fibrations
Date
1995
Author
Ozan, Yıldıray
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If M is a closed smooth manifold, it is well known that M is diffeomorphic to a nonsingular real algebraic set. Let G be a finite group and let X→πY be a principal G-fibration where X and Y are closed smooth manifolds. By the first sentence, we can assume Y is a nonsingular real algebraic set. Question: Is X→πY differentiably equivalent to an algebraic principal G-fibration X~→π~Y (X~, π~ and the action of G on X~ all algebraic)? The author defines an "algebraic cohomology group'' H1A(Y,G) in the case G=(Z/2)k×H, where H is an abelian group of odd order. It is a subgroup of H1(Y,G) and generalizes the usual notion of algebraic cohomology when G=Z/2. The above question is then answered as follows: The existence of X~→π~Y is equivalent to the existence of a regular (= algebraic) classifying map ϕ:Y→K(G,1). Here K(G,1) is an Eilenberg-Mac Lane space which can be realized as the limit of algebraic sets (this uses the special assumption on the type of G) so that algebraicity of the classifying map makes sense. A typical application of the above is the construction of compact nonsingular algebraic sets which admit no nontrivial algebraic G-fibrations.
URI
https://mathscinet.ams.org/mathscinet-getitem?mr=1318735
https://hdl.handle.net/11511/88537
Journal
Contemporary Math
Collections
Department of Mathematics, Article
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Y. Ozan, “Real algebraic principal abelian fibrations,”
Contemporary Math
, 1995, Accessed: 00, 2021. [Online]. Available: https://mathscinet.ams.org/mathscinet-getitem?mr=1318735.