Topological surgery

Topological Surgery on Liouville Tori of an Integrable Hamiltonian System upon... 

It is of importance that manifolds and actually arise in concrete integrable systems. For instance, topological surgery, equivalent to in the dynamics of a three-dimensional heavy rigid body is presented in the work by Kharlamov [1771. 

The general problem is to describe topological surgery on LiouvUle tori occurring at the moment when the point a intersects the bifurcation diagram E. 

If at some point c a continuous curve 7 meets a bifurcation diagram E, then the fibres Ba and Bb may be distinct. If, when moving, the point a punctures E, then the fibre Ba undergoes, generally, a qualitative topological modification, i.e. surgery (Fig. 40). 

In fact, the quadratic IL-spectrum n. is homotopy equivalent to the 0-connective cover of the Quinn (IJ spectrum of simply-connected surgery problems (= topological normal maps... 

By the surgery obstruction realization theorems of Wall 4,SS there exists an (n-q+1)-dimensional topological normal map of triads... 

Proposition 7.3.5 A geometric surgery on a formally n-dlmenslonal topological normal map (f,h) M-- X determines an algebraic surgery on the quadratic kernel o,(f,b). 

The notion of ambient surgery on codimension q submanifolds carries over in the obvious way to such topologic, normal maps. Given a formally (n,n-q)-dimensional topological normal map... 

The LM-gtoups have been used by Cappell and Shaneson (3), and Hambleton [1] to detect which elements of the quadratic L-groups b (Z[n]) of finite groups n ate the surgery obstructio of topological normal maps of closed manifolds (i.e. belong to im (0, (K (Ti, 1) ]Lp)-L (Sfn))). In effect, they... 

Surgery on topological normal maps up to Ep(x)-coefficien homology equivalence was first studied by Jones (1), in... 

Pardon [4] used local surgery theory to extend the work of Madsen, Thomas and Wall on the classification of free acti of finite groups on spheres ("the topological spherical space form problem") to the classification of free actions of finit groups on manifolds which are 3p-homology spheres. 

IfI The computation of odd dimensional projec surgery groups of all finite groups to appear in Topology... 

The Kervaire invariant of framed manifolds and its generalization Ann. of Math. 90, 157 - 186 (1969) f 51 Free Sp-actions on homotopy spheres Proc. 1969 Georgia Conference on the Topology of Manifolds, Markham, 217 - 226 ( [6] Surgery on simply-connected manifolds Springer (1972)... 

A geometric fornmiation of surgery Proc. 1969 Georgia Conference on the Topology of Manifolds, Markham, 500 512 (191 —(TOP) and the surgery obstruction I. 

Proc. 1979 Siegen Topology Symposium, Springer Lecture Notes 788, 482 - 495 (1980) On the obstruction group in homology surgery... 

The total surgery obstruction theory of Ranicki [7] was a tentative first step towards a purely algebraic account of the homotopy theory of compact n-dimensional topological manifolds, at least for n 5, including an algebraic expression for the surgery exact sequence. 

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