Differentialgeometrie und Topologie
Prof. D. Kotschick
Prof. M. Hamilton
Winter term 2016/17
Lecture: Mathematical Gauge Theory II
Exercise sheet 10
1. (Light cone lemma) Let X be a closed oriented 4-manifold with b+
2 (X) = 1. We denote
2
by · the intersection pairing on H (X; R) with values in R. Fix one of the two connected
components of {a ∈ H 2 (X; R) | a2 = a · a > 0} and call this the forward cone. Prove that
the following holds for all elements a, b ∈ H 2 (X; R):
(a) If a is in the forward cone and b in the closure of the forward cone with b 6= 0, then
a · b > 0.
(b) If a and b are in the closure of the forward cone, then a · b ≥ 0.
(c) If a is in the forward cone and b satisfies b2 ≥ 0 and a · b ≥ 0, then b is in the closure
of the forward cone.
2. (Small perturbation Seiberg-Witten invariant) Let X be a closed oriented 4-manifold
−
with b+
2 (X) = 1. Suppose that H1 (X; Z) = 0 and b2 (X) ≤ 9. Assume that X is not spin
−
(by Rokhlin this is automatic if b2 (X) 6= 1).
(a) Let s be a Spinc -structure such that the Seiberg-Witten moduli space has non-negative
expected dimension. Prove that c1 (Ls )2 ≥ 0 and c1 (Ls ) 6= 0.
(b) Choose a forward cone for H 2 (X; R), let g be a Riemannian metric on X and µg the
unique self-dual harmonic 2-form representing a class in the forward cone with µ2g = 1.
For ω ∈ iΩ2+ (X) we define the discriminant by
1
∆Ls (g, ω) =
ω + 2πc1 (Ls ), µg ∈ R.
i
Use the sign of the discriminant to prove that there exists a real number (s, g) > 0
such that there is a well-defined Seiberg-Witten invariant SWXsmall (s) using arbitrary
generic perturbations ω with ||ω|| < (s, g). This Seiberg-Witten invariant is called
small perturbation Seiberg-Witten invariant.
3. (Applications of SWXsmall ) Let X be a closed oriented 4-manifold with b+
2 (X) = 1.
Suppose that H1 (X; Z) = 0, b−
(X)
≤
9
and
X
is
not
spin.
2
(a) Show that there exist only finitely many Spinc -structures s with SWXsmall (s) 6= 0.
(b) Suppose that X admits a Riemannian metric with sg > 0. Prove that SWXsmall ≡ 0.
Please hand in your solutions by Wednesday, January 25, 2016, 12:00 in the box on the
first floor.