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It is well known that compact manifolds of negative sectional curvature don't admit self-maps of degree $>1$. At the same time positively curved manifolds such as $S^n$ and $\mathbb CP^n$ clearly admit self-maps of degree $>1$. Moreover I guess if we take a compact Lie group and consider map $x\to x^2$ the degree of such map is $>1$ (in fact I did not check it).

Question. Let $G$ be a compact Lie group and $H$ its proper connected Lie subgroup. Is it true that $G/H$ admits self-maps of degree $>1$? If not, is there some classification of such quotients and that admit / don't admit a self-map with $deg>1$?

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up vote 4 down vote accepted

If $X$ is a simply connected compact manifold, then one sufficient condition for the existence of maps $X\to X$ of any sufficiently divisible degree is formality: there is a commutative differential graded algebra (cdga) $A$ and cdga maps $A\to H^*(X,\mathbb{R}),A\to \Omega^*(X)$ (the de Rham complex) that both induce isomorphisms in cohomology, see Sullivan, Infinitesimal computations in topology, \S 12, in particular, Theorem 12.2.

Lie groups are formal, and so are their classifying spaces and also many homogeneous spaces, in particular symmetric spaces, see e.g. F\'elix, Halperin, Thomas, Rational homotopy theory, \SS 12 and 15.

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Combining this with weak Lefschetz and formality of compact Kahler manifolds (Deligne-Griffiths-Morgan-Sullivan), this answers the other related question… by the same user. – Donu Arapura Nov 16 '12 at 16:21
Donu -- yes, it does, thanks! – algori Nov 16 '12 at 16:34
By the way, is there any homogeneous space that is not formal? – aglearner Nov 17 '12 at 0:31
aglearner -- yes, there are, see e.g. a recent paper by M. Amann and references therein. – algori Nov 17 '12 at 1:22

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