By Sergei P. Novikov (auth.), S. P. Novikov (eds.)

ISBN-10: 0387170073

ISBN-13: 9780387170077

ISBN-10: 3540170073

ISBN-13: 9783540170075

This booklet constitutes not anything below an up to date survey of the full box of topology (with the exception of "general (set-theoretic) topology"), or, within the phrases of Novikov himself, of what was once termed on the finish of the nineteenth century "Analysis Situs", and consequently different into a few of the subfields of combinatorial, algebraic, differential, homotopic, and geometric topology. The publication offers an outline of those subfields, starting with the weather and continuing correct as much as the current frontiers of analysis. hence one unearths the following the complete variety of topological techniques from fibre areas (Chap.2), CW-complexes, homology and homotopy, via bordism thought and K-theory to the Adams-Novikov spectral series (Chap.3), and in bankruptcy four an exhaustive (but inevitably focused) survey of the speculation of manifolds. An appendix sketching the new awesome advancements within the conception of knots and hyperlinks and low-dimensional topology quite often, brings the survey correct as much as the current. This paintings represents the flagship, because it have been, in whose wake persist with extra specific surveys of a number of the subfields, through quite a few authors.

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Topological Topological spaces. Fibrations. Fibrations. 18 al cover. e. e. to aa homeomorphic to some sorne neighbourhood neighbourhood of the vertex “cross”. On this tree the free group on two generators acts freely in a natural "cross" . this acts a natural way, with orbit space the bouquet S1 V S’. It follows by this sort of argument with orbit space SI SI. It follows 0 that ni(K) is a free group for every 1-complex l-complex (or graph) K. K. / 1 "- \ \ 1 1 \ 1 '\. ...... " ' - - _ . / / Fig.

Fibrations. Fibrations. 7) Important 1. Let Important cases. cases. 1. Let pp :: X X --) Y Y be be aa covering-space covering-space projection projection (see (see above) with (discrete) fiber Fo. Then since ni(Fo, ~0) = 0 for i > (0 denoting denoting above) with (discrete) fiber Fo. (X,Zo) - w(Y, Yo) -5 0; ~o(Fo,Zo) - 0. e. (Le. in in oneoneto-one correspondence with) the number of components of the fiber; in other ta-one correspondence with) the number of components of the fiber; in other 0 words each coset sheet of words each coset corresponds corresponds to to aa sheet of the the covering.

The region U then has as deformation retract the bouquet of the circle and 2-sphere: bouquet of the circle and 2-sphere: u N s2 v s1 = Y. TI(Sl), where infinite cyclic group, with generator t, say. Consider the universal infinite cyclic group, with generator t, say. Consider the universal U, or, rather, the homotopically equivalent universal cover X of the U, or, rather, the homotopically equivalent universal coyer X of the s2 v s1 = Y: S2 V SI = Y: X+Y=S2VS1. 16). 16). Tl (Y) on X is here given by t(X) = x + 1, t (sj2) = s;+1.

### Topology I: General Survey by Sergei P. Novikov (auth.), S. P. Novikov (eds.)

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