Abstract :<br><br>
The Seiberg-Witten invariant is an invariant for closed oriented Spin^c
4-manifold. It is defined to be the number of solutions of the Seiberg-Witten
equation, counted with appropriate sign or multiplicity.
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This definition is valid only when the formal dimension of the moduli space
of solutions is zero.
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When the formal dimension is positive, one possibility to define an invariant
is to use some cohomology classes to evaluate the fundamental class of the
moduli space. The evaluation would give an invariant.
However if you use this method, then the invariant is equal to be zero
for all the known examples of X satisfying b_1(X)=0 and b^+_2(X) >1.
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(cf "Simple type conjecture" :if the formal dimension of the moduli space is
positive, then the invariant of the above type would be always zero for X
with b_1=0 and b^+_2>1)
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There is still another possibility to define an invariant.
One can regard the SW equation itself as an invariant in an appropriate
sense.
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To formulate this idea rigorously we need a finite dimensional approximation
of the SW equation. An alternative way of looking at this is to use a "stable
twisted framing" on the moduli space. The two formulations are related to
each other via a version of Pontrjagin-Thom construction.
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Example:
For any homotopy K3#K3 with spin structure, the formal dimension of the
moduli space is one. We can show that the moduli space has a canonical spin
structure. Our invariant, in this case, is nothing but its spin cobordism
class. It turns out that the class is non-trivial in the degree-1 spin
cobordism group, which is isomorphic to Z/2. A corollary of this is an
adjunction inequality for any homotopy K3#K3.
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Reference
S. Bauer and M. Furuta "A stable cohomotopy refinement of Seiberg-Witen
invariants I" Invent. Math. 155 (2004), 1-19.
<br><br>
M. Furuta and T.J. Li "Pontrjagin-Thom constructions in nonlinear Fredholm
theories", in preparation.