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Algebraic realization of equivariant vector bundles. (English) Zbl 0787.57016
Let $$G$$ be a compact Lie group and $$\Omega$$ an orthogonal representation of $$G$$. We think of an orthogonal representation as an underlying Euclidean space $$\mathbb{R}^ n$$ together with an action of $$G$$ via orthogonal maps. A real algebraic $$G$$ variety is the set of common zeros of polynomials $$p_ 1,\dots,p_ m: \Omega \to \mathbb{R}$$, $V = \{x\in \Omega\mid p_ 1(x) = \cdots = p_ m(x) = 0\},$ which is invariant under the action of $$G$$. Examples of such $$G$$ varieties are the Grassmannians of $$k$$-dimensional subspaces in an $$n$$-dimensional representation $$\Xi$$ of $$G$$ $G_{\mathbb{R}}(\Xi,k) = \{L \in {\mathfrak M}^{n^ 2}_ \mathbb{R}\mid L^ 2 = L,\quad L^ t = L,\quad \text{trace }L = k\}.$ We chose a $$G$$ invariant inner product and an orthonormal basis for $$\Xi$$ so that we could identify the endomorphisms of $$\Xi$$ with real $$n\times n$$ matrices and subspaces of $$\Xi$$ with orthogonal projections onto these subspaces. The action of $$G$$ on $$G_ \mathbb{R}(\Xi,k)$$ is given by conjugation on the endomorphisms of $$\Xi$$. A smooth $$G$$ manifold is said to be algebraically realized if there is an equivariant diffeomorphism $$\varphi: X\to M$$ from a non-singular real algebraic variety $$X$$ to $$M$$. The set of all $$G$$-vector bundles over $$M$$ is said to be algebraically realized by $$X$$ if, for all $$\Xi$$ and $$k$$, every equivariant map $$\mu: X\to G_{\mathbb{R}}(\Xi,k)$$ is equivariantly homotopic to an entire rational map, i.e., every $$G$$ vector bundle over $$X$$ is strongly algebraic. The main result of the paper is
Theorem B. The set of all real $$G$$ vector bundles over a closed smooth $$G$$ manifold is algebraically realized if one of the following assumptions holds:
(1) $$G$$ is the product of a group of odd order and a 2-torus.
(2) The action of $$G$$ on the manifold is semifree.
Let $$Y$$ be a real algebraic $$G$$ variety and $$f: M\to Y$$ an equivariant map. We say that $$(M,f)$$ is algebraically realized by $$(X,\varphi)$$ if $$f\circ \varphi$$ is equivariantly homotopic to an entire rational map. The cobordism class of $$(M,f)$$ has an algebraic representative if $$(M,f)$$ is equivariantly cobordant to a pair $$(Z,\eta)$$ where $$Z$$ is a non- singular real algebraic $$G$$ variety and $$\eta: Z\to Y$$ is entire rational. The proof of theorem B is reduced to a bordism problem.
Theorem C. Let $$G$$ be a compact Lie group. An equivariant map from a closed smooth $$G$$ manifold to a non-singular real algebraic $$G$$ variety is algebraically realized if and only if its equivariant bordism class has an algebraic representative.
Equivariant bordism results of R. E. Stong are then generalized to obtain theorem B [Mem. Am. Math. Soc. 103 (1970; Zbl 0201.255)] and [Duke Math. J. 37, 779-785 (1970; Zbl 0204.236)]. The paper is motivated by the non-equivariant work of J. Nash [Ann. Math., II. Ser. 56, 405-421 (1952; Zbl 0048.385)], A. Tognoli [Ann. Scuola Norm. Sup. Pisa, Sci. fis. mat., III. Ser. 27, 167-185 (1973; Zbl 0263.57011)], S. Akbulut and H. King [Ann. Math., II. Ser. 113, 425-446 (1981; Zbl 0494.57004)], and of R. Benedetti and A. Tognoli [Bull. Sci. Math., II. Sér. 104, 89-112 (1980; Zbl 0421.58001)].
Reviewer: K.H.Dovermann

##### MSC:
 57R85 Equivariant cobordism 57R91 Equivariant algebraic topology of manifolds 14P99 Real algebraic and real-analytic geometry 55R91 Equivariant fiber spaces and bundles in algebraic topology
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