Quotient space

The quotient space /r (0)/G is called a symplectic quotient (or Marsden-Weinstein reduction). It has a complex structure and natural Kahler metric (cf. Theorem 3.30) on points where G acts freely. On the other hand, the set of closed G -orbits is the affine algebro-geometric quotient and denoted by YjjG. In fact, it is known that the above identihcation intertwines the complex structures. 

It is easy to see a matrix has a closed orbit if and only if it is diagonalizable. Hence the set of closed orbits can be identified with the set of eigenvalues. On the other hand, a matrix B with [B, 5I] = 0 (i.e. a normal matrix) can be diagonalizable by a unitary matrix. Hence the quotient space is also identified with the set of eigenvalues. The identification can be seen directly in this example. 

We call this quotient space a hyper-Kahler quotient. 

Since the G-action on is free, the slice theorem implies that the quotient space... 

A simple singularity is a quotient space C /P, where P is a finite subgroup of SU(2). This singularity has been studied from various point of view. See e.g. [5], [76]. 

Show that is an equivalence relation. If the action cr is clear from the context, then the quotient space S/ is often denoted S/G. 

Then the quotient space M/ G (defined in Exercise 4.43) is a differentiable manifold, and the natural projection tt M M/ G is a differentiable function. 

Since the G-action on // 1(C) is free, the slice theorem implies that the quotient space gTl((,)/G has a structure of a C°°-manifold such that the tangent space TaAtt-HO/G) at the orbit G x is isomorphic to the orthogonal complement of Vx in Txg 1( ). Hence the tangent space is the orthogonal complement of Vx IVX JVX KVX in TxX, which is invariant under I, J and K. Thus we have the induced almost hyper-complex structure. The restriction of the Riemannian metric g induces a Riemannian metric on the quotient g 1(()/G. In order to show that these define a hyper-Kahler structure, it is enough to check that the associated Kahler forms u>[, u) 2 and co z are closed by Lemma 3.32. 

The projective plane arises as a quotient space of the sphere, the required group being C,-. It is obtained by identifying antipodal points of the spherical surface in other words, it is the antipodal quotient of the sphere (see Section 1.2.2). P2 is the simplest compact non-orientable surface in the sense that it can be obtained from the sphere by adding just one cross-cap. 

Xq, aside from its generic point, is the quotient space by conjugation ... 

Of course, one easy way to satisfy such a condition is simply to identify all points, and to obtain just a point as the resulting quotient space. The additional requirement for the gluing process is that this identification should be in some sense minimal in other words, no identification is done unless it is a consequence of the prescribed identifications. We shall see how similar universality conditions appear in further definitions of the cell complexes. Furthermore, in Chapter 4 it will be demonstrated how all of these are just special instances of the general colimit construction. 

We also remark that instead of taking the successive joins, one can think of Xi Xfc as the quotient space... 

Definition 6.8. Let f X — Y be a continuous map between two topological spaces. The mapping cylinder of f is the quotient space... 

Definition 15.3. Let T> be a diagram of topological spaces over a trisp A. A colimit ofV is the quotient space colim D = where the... 

The reason for choosing the word torus here is that the quotient space of the plane by this (Z x Z)-action can be viewed as a torus, where each (m, n)-grid path yields a loop following the grid in the northeasterly direction see Figure 20.4. We notice that every torus front has a unique representative starting from the point (0,0). It will soon become clear why we choose to consider the orbits of the action rather than merely considering these representatives of the orbits. 

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