Download e-book for iPad: Algebraic Topology by Arunas Liulevicius (ed.)

By Arunas Liulevicius (ed.)

ISBN-10: 0821814222

ISBN-13: 9780821814222

ISBN-10: 1119672422

ISBN-13: 9781119672425

ISBN-10: 1277292132

ISBN-13: 9781277292138

ISBN-10: 2019691361

ISBN-13: 9782019691363

ISBN-10: 2542682712

ISBN-13: 9782542682715

ISBN-10: 9219661101

ISBN-13: 9789219661103

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To begin with, for the generalisation below, we will not use the assumption that acts freely. The stabiliser of 0 in Aut(D) is the circle, acting in the obvious way. This is just the case β = 0 in the general description above. If is discrete, its intersection with this circle is a discrete subgroup of the circle, which must be a finite cyclic group. So the stabiliser of 0 in is a finite cyclic group Z, say. It follows easily from the discreteness of that we can choose a k < 1 such that if γ ∈ is γ (z) = (αz + β)/(βz + α), where |α|2 − |β|2 = 1, and if |β| ≤ k|β|, then γ is in Z.

Z − z2n ), and where F : X → C is the projection to the z factor. Then = {z1 , . . , z2n } and π1 (C \ ) is generated by 2n loops γ1 , . . , γ2n , where γi is a standard Chapter 4 : Maps between Riemann surfaces loop going once around zi . The degree d is 2, and the representation ρ maps each generator γi to the non-trivial element of S2 (a transposition of the two objects). In traditional language, we make cuts along n disjoint paths joining z2i−1 to z2i for i = 1, . . , n. Then we take two copies of the cut plane and form X \ R by gluing these along the cuts.

E. that for any two points a, b in X+ there are disjoint open sets U, V containing a, b, respectively. If a and b lie in the copy of X0 in X+ , this is clear: we just take corresponding open sets in X0 . So suppose a lies in X0 and b is the point corresponding to 0 in D. Then F0 (a) is not equal to y1 in Y, so we can find a small neighbourhood N of F0 (a) in Y \ which is disjoint from a smaller disc D2 ⊂ D1 containing y1 . The open sets F0−1 (N) and {0} ∪ φ −1 (F0−1 (D2 )) are disjoint in X+ and contain b, a, respectively.

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Algebraic Topology by Arunas Liulevicius (ed.)

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