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Lectures on Geometric Group Theory

FIBRED COARSE EMBEDDINGS, A-T
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... constructive, giving a method for defining an ordered decomposition space, and, consequently, an ordered quotient space. Also presented is a definition for an ordered quotient map which fits neatly into the rest of the theory. We begin by reviewing pertinent notions from the theory of ordered topologi ...
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...  ( a , t )  a , a  A, t [ 0 ,1] [10,12,13,15,18] . A map : M   N , where M and N are c  –Riemannian manifolds of dimension m , n respectively is said to be an isometric folding of M into N , iff for any piecewise geodesic path  : J   M , the induced path    : J   N is a piecewise ...
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... So we can formulate continuity purely in terms of open sets, banishing those s and δs completely. That is the basis of topology, but we would like to apply it to more than just functions on the real line. In order to define continuity for functions between any sets, we have seen that we only need t ...
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... on the categorical Tychonoff Theorem [7] that had already been used to affirm product stability of c-compactness for topological groups: see Example 9.5 of [5]. In fact, we not only extend but slightly generalize the known object-level results since, unlike the authors of [12, 19] and of most papers ...
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Fundamental group

In the mathematics of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a topological invariant: homeomorphic topological spaces have the same fundamental group.Fundamental groups can be studied using the theory of covering spaces, since a fundamental group coincides with the group of deck transformations of the associated universal covering space. The abelianization of the fundamental group can be identified with the first homology group of the space. When the topological space is homeomorphic to a simplicial complex, its fundamental group can be described explicitly in terms of generators and relations.Henri Poincaré defined the fundamental group in 1895 in his paper ""Analysis situs"". The concept emerged in the theory of Riemann surfaces, in the work of Bernhard Riemann, Poincaré, and Felix Klein. It describes the monodromy properties of complex-valued functions, as well as providing a complete topological classification of closed surfaces.
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