## Nilpotency, almost nonnegative curvature, and the gradient flow on Alexandrov spaces.(English)Zbl 1192.53040

A closed smooth manifold $$M$$ of dimension $$m$$ is said to be almost nonnegatively curved if it can Gromov-Hausdorff converge to a single point under a lower curvature bound, i.e., if it admits a sequence of Riemannian metrics with sectional curvatures bounded from below by $$-1/n$$ and diameters bounded from above by $$1/n$$. The authors show:
Theorem A. Let $$M$$ be a closed almost nonnegatively curved manifold. Then a finite cover of $$M$$ is a nilpotent space. The authors also provide an affirmative answer to a conjecture of K. Fukaya and T. Yamaguchi [Ann. Math. (2) 136, No. 2, 253–333 (1992; Zbl 0770.53028)].
Theorem B. Let $$M$$ be an almost nonnegatively curved $$m$$-manifold. Then $$\pi_1(M)$$ contains a nilpotent subgroup of index at most $$C(m)$$. The authors also establish a fibration theorem.
Theorem C. Let $$M$$ be an almost nonnegatively curved manifold. Then a finite cover $$\tilde M$$ of $$M$$ is the total space of a fiber bundle $$F\rightarrow\tilde M\rightarrow N$$ over a nilmanifold $$N$$ with simply connected fiber $$F$$. Moreover, the fiber $$F$$ is almost nonnegatively curved in a generalized sense.
The authors use Alexandrov geometry and the gradient flow of the square of a distance function. They use “limit fundamental groups of Alexandrov spaces” and discuss some open questions.

### MSC:

 53C20 Global Riemannian geometry, including pinching

Zbl 0770.53028
Full Text:

### References:

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