Algebraic realization of equivariant vector bundles.

*(English)*Zbl 0787.57016Let \(G\) 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)].

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 |