Some articles on contractible:
Classifying Space - Motivation
... another way to specify the condition on X is that the universal cover Y of X is contractible ... higher homotopy groups vanish the fundamental idea is, given G, to find such a contractible space Y on which G acts freely ... remark that an infinite cyclic group C acts freely on the real line R, which is contractible ...
... another way to specify the condition on X is that the universal cover Y of X is contractible ... higher homotopy groups vanish the fundamental idea is, given G, to find such a contractible space Y on which G acts freely ... remark that an infinite cyclic group C acts freely on the real line R, which is contractible ...
CAT(k) Space - Properties of CAT(k) Spaces
... The d-balls in X of radius less than Dk are contractible ... these properties that, for k ≤ 0, the universal cover of every CAT(k) space is contractible in particular, the higher homotopy groups of such a space are trivial ... example of the n-sphere Sn shows, there is, in general, no hope for a CAT(k) space to be contractible if k is strictly positive ...
... The d-balls in X of radius less than Dk are contractible ... these properties that, for k ≤ 0, the universal cover of every CAT(k) space is contractible in particular, the higher homotopy groups of such a space are trivial ... example of the n-sphere Sn shows, there is, in general, no hope for a CAT(k) space to be contractible if k is strictly positive ...
Locally Contractible Spaces
... A topological space is locally contractible if every point has a local base of contractible neighborhoods ... Contractible spaces are not necessarily locally contractible nor vice-versa ... For example, the comb space is contractible but not locally contractible (if it were, it would be locally connected which it is not) ...
... A topological space is locally contractible if every point has a local base of contractible neighborhoods ... Contractible spaces are not necessarily locally contractible nor vice-versa ... For example, the comb space is contractible but not locally contractible (if it were, it would be locally connected which it is not) ...
Projective Unitary Group - The Topology of PU(H) - PU(H) Is A Classifying Space For Circle Bundles
... This means that U(H) is weakly contractible, and an additional argument shows that it is actually contractible ... finite-dimensional cousins U(n) and their limit U(∞) under the inclusion maps which are not contractible admitting homotopically nontrivial continuous mappings onto U(1) given by the determinant of matrices ... Thus U is a contractible space with a U(1) action, which identifies it as EU(1) and the space of U(1) orbits as BU(1), the classifying space for U(1) ...
... This means that U(H) is weakly contractible, and an additional argument shows that it is actually contractible ... finite-dimensional cousins U(n) and their limit U(∞) under the inclusion maps which are not contractible admitting homotopically nontrivial continuous mappings onto U(1) given by the determinant of matrices ... Thus U is a contractible space with a U(1) action, which identifies it as EU(1) and the space of U(1) orbits as BU(1), the classifying space for U(1) ...
Contractible Space - Examples and Counterexamples
... Any Euclidean space is contractible, as is any star domain on a Euclidean space ... The Whitehead manifold is contractible ... Spheres of any finite dimension are not contractible ...
... Any Euclidean space is contractible, as is any star domain on a Euclidean space ... The Whitehead manifold is contractible ... Spheres of any finite dimension are not contractible ...
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