Summary: REAL ALGEBRAIC STRUCTURES
Abstract. A brief survey of real algebraic structures on topological spaces is given.
The question of when a manifold M is homeomorphic (or diffeomorphic) to a real
algebraic set V is an old one. If we start with an imbedding M Rn
on finding an algebraic subset V of Rn
which is isotopic to M in Rn
, the problem
encounters additional difficulties coming from complexification. Hence it is natural to
break the question into two parts: (1) Stable: If M homeomorphic (or diffeomorphic)
to some real algebraic set. (2) Ambient: If M isotopic to a real algebraic subset in Rn
While the first problem has a complete solution, the second one has many interesting
obstructions. Here we give a quick summary of some of the related results. Clearly
this brief survey is by no means complete, and it is biased towards author's interest
in the field. For the basics reader can consult to the book [AK3].
1. Stable Results
By [N] and [T] every closed smooth manifold is diffeomorphic to a nonsingular real