On Colimits in Various Categories of Manifolds
... This map is continuous and descends to a map RP3 /G → C(RP2 ) because negating coordinates does not matter (because all coordinates get squared). So we have a continuous bijection between compact Hausdorff spaces, i.e. a homeomorphism. However, C(RP2 ) is not a topological manifold (with or without ...
... This map is continuous and descends to a map RP3 /G → C(RP2 ) because negating coordinates does not matter (because all coordinates get squared). So we have a continuous bijection between compact Hausdorff spaces, i.e. a homeomorphism. However, C(RP2 ) is not a topological manifold (with or without ...
Finite Spaces Handouts 1
... Assume we have an equivalence relation ∼ on a topological space (X, U). Let X/ ∼ be the set of equivalence classes. There is a natural map q : X → X/ ∼, which sends each element to its equivalence class. Clearly q is surjective, so we can talk about the quotient topology on X/ ∼. Fun Exercise 1.4.4. ...
... Assume we have an equivalence relation ∼ on a topological space (X, U). Let X/ ∼ be the set of equivalence classes. There is a natural map q : X → X/ ∼, which sends each element to its equivalence class. Clearly q is surjective, so we can talk about the quotient topology on X/ ∼. Fun Exercise 1.4.4. ...
A few results about topological types
... Definition 16 ((topological) neighbourhood system) Let (U, TU ) be a topological space and x ∈ U ; the neighbourhood system of x, noted ν(x), is the set of all its neighbourhoods. 2 Definition 17 ((topological) neighbourhood base) Let (U, TU ) be a topological space and x ∈ U ; a neighbourhood base ...
... Definition 16 ((topological) neighbourhood system) Let (U, TU ) be a topological space and x ∈ U ; the neighbourhood system of x, noted ν(x), is the set of all its neighbourhoods. 2 Definition 17 ((topological) neighbourhood base) Let (U, TU ) be a topological space and x ∈ U ; a neighbourhood base ...
Topology I
... Let U be an open subset of . For x, y ∈ U, set x ≡ y iff [x, y] ∪ [y, x] ⊆ U. It can be easily checked that this is an equivalence relation. The equivalence classes (which form a partition of U) must be open intervals (because U is open). Since each open interval can be coded by one of the rational ...
... Let U be an open subset of . For x, y ∈ U, set x ≡ y iff [x, y] ∪ [y, x] ⊆ U. It can be easily checked that this is an equivalence relation. The equivalence classes (which form a partition of U) must be open intervals (because U is open). Since each open interval can be coded by one of the rational ...
Homotopy type of symplectomorphism groups of × S Geometry & Topology
... identity on H2 (S 2 × S 2 ; Z). Gromov proved that G0 is connected and it is homotopy equivalent to its subgroup of standard isometries SO(3) × SO(3). He also showed that this would no longer hold when one sphere is larger than the other, and in [9] McDuff constructed explicitly an element of infini ...
... identity on H2 (S 2 × S 2 ; Z). Gromov proved that G0 is connected and it is homotopy equivalent to its subgroup of standard isometries SO(3) × SO(3). He also showed that this would no longer hold when one sphere is larger than the other, and in [9] McDuff constructed explicitly an element of infini ...
Orientability
In mathematics, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a ""clockwise"" direction of loops in the surface, as needed by Stokes' theorem for instance. More generally, orientability of an abstract surface, or manifold, measures whether one can consistently choose a ""clockwise"" orientation for all loops in the manifold. Equivalently, a surface is orientable if a two-dimensional figure such as 20px in the space cannot be moved (continuously) around the space and back to where it started so that it looks like its own mirror image 20px.The notion of orientability can be generalised to higher-dimensional manifolds as well. A manifold is orientable if it has a consistent choice of orientation, and a connected orientable manifold has exactly two different possible orientations. In this setting, various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds often employ methods of homology theory, whereas for differentiable manifolds more structure is present, allowing a formulation in terms of differential forms. An important generalization of the notion of orientability of a space is that of orientability of a family of spaces parameterized by some other space (a fiber bundle) for which an orientation must be selected in each of the spaces which varies continuously with respect to changes in the parameter values.