1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

Lecture 1. Background from Differential Geometry The data for the Seiberg-Witten equations are a connection A on a certain line ...

108 M. HUTCHINGS AND C. H. TAUBES, SEIBERG-WITTEN EQUATIONS It is a consequence of the definition that the fiber Ex of E at any ...

LECTURE 1. BACKGROUND FROM DIFFERENTIAL GEOMETRY '109 certain compatiblity conditions, we call the splitting a connection. (Such ...

110 M. HUTCHINGS AND C. H. TAUBES, SEIBERG-WITTEN EQUATIONS If Aab is the matrix for A in a local trivialization as in (1.2), an ...

LECTURE 1. BACKGROUND FROM DIFFERENTIAL GEOMETRY 111 is defined as follows. If v , w are two vectors in T xX, extend them to loc ...

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LECTURE 2 Spin and the Seiberg-Witten Equations As we said at the beginning of the last lecture, the data for the Seiberg-Witten ...

114 M. HUTCHINGS AND C. H. TAUBES, SEIBERG-WITTEN EQUATIONS associated vector bundle Vp =(P x V)/ "', (pg, v) rv (p, p(g)v). Exe ...

LECTURE 2. SPIN AND THE SEIBERG-WITTEN EQUAT IONS 115 Theorem 2.5. On any oriented 4-manifold X, Spine structures exist, and the ...

116 M. HUTCHINGS AND C. H. TAUBES, SEIBERG-WITTEN EQUATIONS and let h E SU(2) act on x E JR^3 by h·x=hxh-^1. Exercise 2.7. Check ...

LECTURE 2. SPIN AND THE SEIBERG-WITTEN EQUATIONS 117 Remark 2.10. A Spine structure is actually equivalent to the bundle S = S_ ...

118 M. HUTCHINGS AND C. H. TAUBES, SEIBERG-WITTEN EQUATIONS acting on C^2 -valued functions. (The factor of ~ co mes from the sq ...

LECTURE 2. SPIN AND THE SEIBERG-WITTEN EQUATIONS 119 Let's work out the second of the Seiberg-Witten equations in our local mode ...

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Lecture 3. The Seiberg-Witten Invariants Let X be a closed oriented smooth 4-manifold. Let g be a Riemannian metric on X and let ...

122 M. HUTCHINGS AND C. H. TAUBES, SEIBERG-WITTEN EQUATIONS Here EX is some base point. The group{¢ E C^00 (X,S^1 ): ¢() = 1} of ...

LECTURE 3. THE SEIBERG-WITTEN INVARIANTS 123 standard package for dealing with equations of this type. Another example of an ell ...

124 M. HUTCHINGS AND C. H. TAUBES, SEIBERG-WITTEN EQUATIONS is a tensor with components R~, antisymmetric in (i, j) and also in ...

LECTURE 3. THE SEIBERG-WITTEN INVARIANTS 125 where V is a certain linear combination of derivatives. By this equation, the bound ...

126 M. HUTCHINGS AND C. H. TAUBES, SEIBERG-WITTEN EQUATIONS The invariants bi, b^1 are additive under this operation. Note that ...