Topology of Injective Endomorphisms of Real Algebraic Sets

Using only basic topological properties of real algebraic sets and regular morphisms we show that any injective regular self-mapping of a real algebraic set is surjective. Then we show that injective morphisms between germs of real algebraic sets define a partial order on the equivalence classes of these germs divided by continuous semi-algebraic homeomorphisms. We use this observation to deduce that any injective regular self-mapping of a real algebraic set is a homeomorphism. We show also a similar local property. All our results can be extended to arc-symmetric semi-algebraic sets and injective continuous arc-symmetric morphisms, and some results to Euler semi-algebraic sets and injective continuous semi-algebraic morphisms. In 1960 Newman [16] showed that any injective real polynomial map R2 → R2 is surjective. This was extended to the real polynomial maps R → R in 1962 by Bia lynicki-Birula and Rosenlicht [5]. In 1969 Ax [3] showed that any injective regular self-mapping of a complex algebraic variety is surjective. Unilike the proof of [5], that was topological, the proof of Ax is based on the Lefschetz principle and a reduction to the finite field case. In [7] (1969) Borel extended the idea of [5] and gave a topological proof of Ax’ Theorem that can be applied to real algebraic non-singular varieties. The first proof of an analogous result for singular real algebraic varieties was given in 1999 by Kurdyka [11]. Kurdyka’s proof is based on Borel’s argument and the geometry of semi-algebraic arc-symmetric sets. For more on history of the problem of surjectivity of injective mappings, the motivation, and a wide spectrum of possible applications we refer the reader to a recent paper of Gromov [9]. We note that Borel’s proof gives as well that an injective regular self-mapping of a complex algebraic variety or of a non-singular real algebraic variety is a homeomorphism. The analogous statement in the general real algebraic case has not been proven till now (to the author’s best knowledge) and doesn’t not follow from [11]. In this paper we first present a new topological proof of Kurdyka’s theorem and then extend the argument to show that injective regular self-mappings of real algebraic varieties are homeomorphisms. Moreover we obtain local versions of this result and establish a hierarchy of real algebraic singularities (more generally of Euler semi-algebraic singularities) with respect to injective regular mappings. Our approach is based on two classical topological properties of real algebraic sets and maps. The first one is Sullivan’s Theorem that each real algebraic set is Euler (see definition 1.1 below). The second one says that for a regular map f : X → Y of real algebraic sets there exists a proper real algebraic subset Y ′ ⊂ Y such that the Euler characteristic of the fibres of f , taken modulo 2, is constant on Y \ Y ′. This is the crucial observation for the problem since it implies that the image of injective regular map of 1991 Mathematics Subject Classification. 14Pxx, 14A10, 32B10.