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New invariants of 3- and 4-dimensional manifolds. (English) Zbl 0667.57018
The mathematical heritage of Hermann Weyl, Proc. Symp., Durham/NC 1987, Proc. Symp. Pure Math. 48, 285-299 (1988).
[For the entire collection see Zbl 0644.00001.]
The Casson invariant \(\lambda\) (Y) of an oriented homology 3-sphere Y is roughly defined by \(2\lambda (Y)=\{number\) of irreducible representations \(\pi_ 1(Y)\to SU(2)\}\). It can be viewed as the Euler characteristic \(\sum^{7}_{q=0}(-1)^ q \dim HF^+_ q(Y)\) of the Witten-Morse complex \(C_*\) for the Chern-Simons function f: \({\mathcal C}\to R/Z\), where the infinite-dimensional manifold \({\mathcal C}\) is the space of (classes of) SU(2)-connections for the trivial bundle over Y. HF stands for Floer homology. There is a deep connection between the Donaldson invariants \(\Phi_ k\) of an indecomposable 4-manifold W having splittable intersection form, and Floer homology of a homology 3-sphere \(Y\subset W\) which realizes the splitting. Not all of the above results are fully written down, but the general picture seems to be fairly clear and the ideas are so bautiful and simple that they deserve a nontechnical presentation. The author fulfils such a presentation in a brillant manner.
Reviewer: S.V.Matveev

MSC:
57R55 Differentiable structures in differential topology
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
81T17 Renormalization group methods applied to problems in quantum field theory
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
57N10 Topology of general \(3\)-manifolds (MSC2010)