By S. K. Donaldson

This monograph supplies an intensive exposition of Floer's seminal paintings in the course of the Eighties from a latest standpoint. the fabric contained the following was once built with particular purposes in brain. even though, it has now develop into transparent that the strategies used are vital for lots of present parts of study. a massive instance will be symplectic conception and gluing difficulties for self-dual metrics and different metrics with certain holonomy. the writer writes with the massive photo continuously in brain. in addition to a evaluate of the present kingdom of information, there are sections at the most likely course of destiny examine. integrated during this are connections among Floer teams and the distinguished Seiberg-Witten invariants. the implications defined during this quantity shape a part of the world referred to as Donaldson idea. the importance of this paintings is such that the writer was once provided the distinguished Fields Medal for his contribution.

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**Example text**

1]), and the ideas can be extended to reducibles (we need not worry about potential technical diﬃculties here since we need the result only for motivation). The horizontal condition can be expressed directly in terms of the connection in four dimensions. We say a connection A on π ∗ (P ) is in temporal gauge if the covariant derivative of any section lifted up from Y along d in the R-direction vanishes. 9: ∂ At = it (FA ), ∂t d , and At is the family of where it denotes contraction by the vector dt connections on P obtained by restriction to slices.

It follows then from the argument above that DA is Fredholm if a = a1 + a2 where a1 ∈ C0 and a2 ∈ L4 . 3 The addition property The index invariant deﬁned in the previous Section has a simple formal property which is basic to Floer’s theory. Let A be an adapted connection (with acyclic limits) on a bundle P over a 4-manifold X with tubular ends, as considered above, and suppose that X contains two boundary components Y, Y , where Y is isometric to Y with the reversed orientation. More precisely, X has an end Y × (0, ∞) and another end Y × (0, ∞).

2 The index 51 D then it follows immediately from the deﬁnition of the formal adjoint that D∗ (g) = 0. 1. Thus g is in the orthogonal complement of the image of D and we have shown that (Im D)⊥ = ker D∗ ∩ L2 . One should notice here one diﬀerence between our set-up and the usual compact case: the failure of a ‘Rellich lemma’ in our function spaces. That is, the embedding L21 → L2 is not compact. To see this one need only consider the translates of a compactly supported section over the tube. Of course, in the usual case this Rellich lemma gives immediately the ﬁnite dimensionality of ker D.