By Allan J. Sieradski
The therapy of the topic of this article isn't encyclopedic, nor used to be it designed to be appropriate as a reference guide for specialists. relatively, it introduces the subjects slowly of their historical demeanour, in order that scholars aren't beaten by means of the last word achievements of a number of generations of mathematicians. cautious readers will see how topologists have progressively subtle and prolonged the paintings in their predecessors and the way such a lot stable rules succeed in past what their originators estimated. To inspire the improvement of topological instinct, the textual content is abundantly illustrated. Examples, too a variety of to be thoroughly coated in semesters of lectures, make this article compatible for self sufficient research and make allowance teachers the liberty to choose what they'll emphasize. the 1st 8 chapters are appropriate for a one-semester path often topology. the whole textual content is appropriate for a year-long undergraduate or graduate point curse, and gives a powerful origin for a next algebraic topology path dedicated to the better homotopy teams, homology, and cohomology.
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Additional info for An Introduction to Topology and Homotopy
Since the surgery obstruction also vanishes we can even find a free action by Q on a homology 3-sphere, whether this can be taken simply connected should be regarded as an open question. The argument outlined in [Th 21 has a gap, the filling of which depends either on extending work of H. Miller to describe the unstable homotopy set [BQ, BSO(4)] or on an argument with KSp-valued characteristic classes to exclude the stable classes defined by certain virtual representations from this set. I have yet to complete the latter.
Since Aut(C t) has order 2t-1 the cyclic case is trivial, for the two 2 others note that Aut(H) only contains elements of odd order if H = C2 X C2 or D*8. In both cases p = 3, and the non-trivial extensions are PSL(2,1F3) and SL(2,F3) respectively. p'-groups we have proved that G/A is G isomorphic to one of the groups C2t, D* t, T*1 or 0*1, but before giving the classification in its most useful form we must allow for the possibility that 3 divides the order of AG in the last two cases. Thus we must describe all extensions of C3v-1 by either T*1 or 0*1 which satisfy the condition.
Inductively we may suppose that all such have been removed. Consider the one remaining curve C1 in K n AK. be a small tubular neighbourhood of C1 such that N Let N n AN = 0. Write C. R = M - N - (AN) L (A2N)u , so that R - K - AK - A2K = R1 U R2 (disjoint union) when R1 and R2 are A-invariant solid tori. The orbit space is 47 a union of 3 solid tori, which intersect in pairs along boundary annuli. 1 this is enough to establish the existence of a Seifert fibration (with base space S2 and at most 3 exceptional fibres).
An Introduction to Topology and Homotopy by Allan J. Sieradski