Download e-book for iPad: Algebraic and Geometric Topology, Part 2 by Milgram R. (ed.)

By Milgram R. (ed.)

ISBN-10: 0821814338

ISBN-13: 9780821814338

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Additional resources for Algebraic and Geometric Topology, Part 2

Sample text

3. The minimum value of the maximum m(Et) in (2) taken 51 over all one-parameter families is 6. 2 carefully to reduce the maximum for a family £t if this exceeds 12. (i) This is done by noting there must be a reasonable number of available 2-gons for each non-critical value of t, since one is dealing with three simple surfaces. (ii) One must choose the 2-gons away from where critical points of the intersection of the three surfaces are about to occur, so as to continue with the family £t after modification.

To a free action of T*1 on S3. Only the last T corresponds For 0*1 the argument is similar (and helped by the fact that T*1 is a subgroup of index 2); there are two irreducible two dimensional complex representations 0+ with no eigenvalue equal to 1. These may be distinguished by their restrictions to D* 16' for which we obtain the two representations which restrict in their turn to the standard inclusion of D*8 = {+ 1, + i, + j, + k} in Sp(1) = SU(2). The group T* v (v ? 2) is a little more interesting - if we use the isomorphism of Spin(4) with SU(2) x SU(2) mentioned at the start of this chapter, and recall that T* v contains an index 3 subgroup isomorphic to C v-1 x D*8, we see that any monomorphism into SO(4) must map the 3-Sylow 3 subgroup generated by X to (C k+3v-1 ® -k+3v-1 ), where r, is a primitive v 3-th root of unity and (k,3v = 1.

If y is isotopic to the core of either "handle" in M, then M* inherits a similar splitting from M and is a lens space. Otherwise write M-N = V 0 u V1 , where F identifies F part of the boundaries of two solid tori along an annulus (T-N n T). mi ? 2 The central loop in F is homologous to multiples of the cores of the V. (i = 1,0), so that M-N is 37 Seifert fibered with exceptional fibres of multiplities m0 and mi over an orbit space D2. Seifert fibered. It follows that M*-N* is also (This step is not trivial.

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Algebraic and Geometric Topology, Part 2 by Milgram R. (ed.)


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