Quadratic enhancements of surfaces: two vanishing results

This note records two results which were inexplicably omitted from our paper on Pin structures on low dimensional manifolds, [KT]. Kirby chose not to be listed as a coauthor. A Pin^- structure on a surface F induces a quadratic enhancement of the mod 2 intersection form, q: H_1(F;Z/2Z) ->Z/4Z Theorem 1.1 says that q vanishes on the kernel of the map in homology to a bounding 3-manifold. This is used by Kreck and Puppe (arXiv:0707.1599 [math.AT]) who refer for a proof to an email of the author to Kreck. A more polished and public proof seems desirable. In [KT], section 6, a Pin^- structure is constructed on a surface F dual to w_2 in an oriented 4-manifold M^4. Theorem 2.1 says that q vanishes on the Poincare dual to the image of H^1(M^4;Z/2Z) in H^1(F;Z/2Z).

Here x • y denotes the mod-2 intersection of the two classes and 2 · : Z/2Z → Z/4Z denotes the standard inclusion. Conversely, every such function comes from a unique P in − -structure. This is classical, but see [1, Section 3]. Theorem 1.1. Let M 3 be a 3-manifold with a fixed P in − -structure and let F be the boundary of M . Give F the induced P in − -structure. Let x ∈ H 1 (F ; Z/2Z) be a class which vanishes in H 1 (M ; Z/2Z). Then q(x) = 0.
Proof. We start with two lemmas. Lemma 1.2. Let S 1 ⊂ F be an embedded circle with trivial normal bundle. Fix a P in − -structure on F . One can do surgery on this embedding and extend the P in − -structure to the trace of the surgery if and only if q(S 1 ) = 0.
Proof. Pick a point on the circle and orient the tangent space at this point and also orient the circle at the point. This orients, and hence frames, the normal bundle to S 1 in F .
A tubular neighborhood of the circle is now oriented and so a P in − -structure on F restricts to a Spin-structure on this neighborhood. The framing on the normal bundle induces a stable framing of S 1 and q(S 1 ) ∈ MSpin 1 ∼ = Z/2Z. This is equivalent to the description in Kirby-Taylor [1] just before Definition 3.5. The definition given there works for all circles, not just ones with trivial normal bundle.
The trace of the surgery is formed by gluing D 2 × D 1 to S 1 × D 1 . Since 1-dimensional framed bordism is Z/2Z and maps isomorphically to MSpin 1 , if q(S 1 ) = 0 ∈ MSpin 1 then the Spin-structure on the circle extends over a disk and hence over D 2 × D 1 and finally over the entire trace.
The only if part follows from the next lemma.
Lemma 1.3. Let X be a surface bounding a 3-manifold W and suppose W has a P in − -structure. Let S 1 ⊂ F be an embedded circle which bounds an embedded disk in W . Then q(S 1 ) = 0.
Proof. A tubular neighborhood of the disk in W is trivial. As in the proof of the first lemma, orient a point on the circle and orient the circle. These orientations extend over the neighborhood of the disk and over the disk and hence frame the normal bundle of the disk in W . The P in − -structure on W restricts to a Spin structure on the neighborhood of the disk. Lemma 2.7 of [1] shows that restricting to the disk and then to the bounding circle gives the same Spin-structure as restricting to X and then to the circle. The first restriction is obviously 0 and the second is q(S 1 ).
Turn now to the proof of the Theorem.
Proof. Assume M is connected and hence has a handlebody decomposition with no 0-handles. Divide M into two pieces, Y and M ′ . The submanifold Y is obtained from F by attaching the 1-handles. Let X be the rest of the boundary of Y . Let M ′ be the result of attaching the 2 and 3 handles to X so that M = Y ∪ M ′ . Notice that Y is obtained from X by adding 2-handles. The P in − -structure on M restricts to one on Y and one on M ′ and hence one on X. Let q X denote the resulting quadratic enhancement.
Let x ∈ H 1 (F ; Z/2Z) be a class that vanishes in H 1 (M ; Z/2Z). The class x can always be represented by disjoint embedded circles using the usual trick for removing transverse intersections. Let κ ∈ H 2 (M, F ; Z/2Z) be the resulting relative class. By excision, H 2 (M, F ; Z/2Z) = H 2 (M ′ , X; Z/2Z) so let x 1 ∈ H 1 (X; Z/2Z) be the boundary of κ after excision. It follows from the construction that x 1 vanishes in H 1 (M ′ ; Z/2Z) and that there is a relative class λ ∈ H 2 (Y, F ⊥ ⊥ X; Z/2Z) with boundary x + x 1 .
Every homology class in X which dies in M ′ can be represented after handle slides and additions by the boundary of a 2-handle. But this is a surgery so by Lemma 1.2, q X (x 1 ) = 0.
By adding all the 1-handles in a small disk in F , we see X as a connected sum of F and some tori and Klein bottles. In particular, there is a classx in H 1 (X; Z/2Z) which can be joined to x ∈ H 1 (F ; Z/2Z) by an embedded cylinder in Y . Then x + x 1 vanishes in H 1 (Y ; Z/2Z) and again Y has no 0 or 1-handles when built from X, so q X (x + x 1 ) = 0.
Make the cylinder from x tox transverse to a surface spanning x and x 1 representing λ. The intersection will be some circles and some arcs and the usual "arc has two ends" argument showsx • x 1 = x • x. But since x bounds in M , x • x = 0. Hence 0 = q X (x + x 1 ) = q X (x) + q X (x 1 ) = q X (x).
Since we added the 1-handles in a small disk D 2 ⊂ F , we see an embedding Furthermore, this embedding is the inclusion F ′ ⊂ F at F ′ × 0 and is the inclusion F ′ ⊂ X at F ′ × 1. Here F ′ ⊂ X is the inclusion whose complement is the connected sum of tori and Klein bottles. The cylinder between our representatives of x andx lies in F ′ × [0, 1].
Since the Spin-structure on the circles representing x orx can be computed from the P in − -structure on a neighborhood of these circles [1, Definition 3.5], we can work in F ′ × [0, 1] with its induced P in − -structure.
The P in − -structures on F ′ × 0 and F ′ × 1 are equivalent and since equivalent P in − -structures induce the same restriction to an S 1 ⊂ F ′ , q X (x) = q(x) and hence q(x) = 0.

The dual to w 2
Let M 4 be an oriented 4-manifold and let F ⊂ M be dual to w 2 . In [1, Section 6] we used a Spin-structure on M − F to construct a P in − -structure on F .