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If a compact manifold M has nonnegative Ricci curvature tensor, then its fundamental group has at most polynomial growth. On the other hand, if M has negative curvature, then ...

A collection of open sets of a topological space whose union contains a given subset. For example, an open cover of the real line, with respect to the Euclidean topology, is ...

An open interval is an interval that does not include its end points. The open interval {x:a<x<b} is denoted (a,b), although the nonstandard notation ]a,b[ is sometimes also ...

A manifold is said to be orientable if it can be given an orientation. Note the distinction between an "orientable manifold" and an "oriented manifold," where the former ...

The Pontryagin number is defined in terms of the Pontryagin class of a manifold as follows. For any collection of Pontryagin classes such that their cup product has the same ...

An iterated fibration of Eilenberg-Mac lane spaces. Every topological space has this homotopy type.

A subspace A of X is called a retract of X if there is a continuous map f:X->X (called a retraction) such that for all x in X and all a in A, 1. f(x) in A, and 2. f(a)=a. ...

A retraction is a continuous map of a space onto a subspace leaving each point of the subspace fixed. Alternatively, retraction can refer to withdrawal of a paper containing ...

If J is a simple closed curve in R^2, the closure of one of the components of R^2-J is homeomorphic with the unit 2-ball. This theorem may be proved using the Riemann mapping ...

Let H be a Hilbert space, B(H) the set of bounded linear operators from H to itself, T an operator on H, and sigma(T) the operator spectrum of T. Then if T in B(H) and T is ...