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Rational visibility of a Lie group in the monoid of self-homotopy equivalences of a homogeneous space. (English) Zbl 1247.55006

This paper deals with the concept of rational visibility when applied to a particularly interesting situation. Given a map \(f: X\to Y\) between, say, simply connected spaces, \(X\) is said to be rationally visible if \(\pi_*(f)\otimes {\mathbb Q}\) is injective. If \(G\) is a Lie group and \(M\) is a homogeneous space admitting a left translation \(G\times M\to M\), it is of interest to determine the degree of rational visibility of the map \(\lambda: G\to\)aut\(_1(M)\), \(\lambda(g)(x)=gx\). Here, aut\(_1(M)\) denotes the monoid of self equivalences of \(M\) homotopic to the identity.
The main result of general nature contains criteria for rational visibility in a special instance of this: if \(U\) denotes a simply connected closed subgroup of the simply connected Lie group \(G\), the visibility of certain rational homotopy groups of aut\(_1(G/U)\) is enabled by cohomological properties of the fibration \(G/U\to BU\to BG\).
From this, the author obtains interesting results on the rational homotopy behavior of various, in principle rigid, structures.
For instance, if \(M\) is a homogenous space admitting a left translation by the Lie group \(G\), then \(\pi_n(\)Diff\(_1M)\) contains an element of infinite order as long as \(G\) is visible in aut\(_1M\) in degree \(n\).
Another consequence is a former result of Kedra, McDuff and Sasao: \(SU(m)\) is rationally visible in the flag manifold \(U(m)/U(m_1)\times\dots\times U(m_n)\).
Many other examples of rational visibility of classical compact Lie groups on classical homogeneous spaces are given.
The main tool to achieve these results is the detailed study of the rational homotopy groups of the mapping spaces involved through their algebraic Sullivan models.

MSC:

55P62 Rational homotopy theory
57R19 Algebraic topology on manifolds and differential topology
57R20 Characteristic classes and numbers in differential topology
57T35 Applications of Eilenberg-Moore spectral sequences
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