Unfolded Seiberg-Witten Floer spectra. II: Relative invariants and the gluing theorem. (English) Zbl 1540.57028

Khandhawit, Tirasan; Lin, Jianfeng; Sasahira, Hirofumi

J. Differ. Geom. 124, No. 2, 231-316 (2023).

The authors extend relative Bauer-Furuta invariants [S. Bauer and M. Furuta, Invent. Math. 155, No. 1, 1–19 (2004; Zbl 1050.57024)] to general 4-manifolds with boundary using the construction of unfolded Seiberg-Witten-Floer spectra for general 3-manifolds contained in their earlier paper [T. Khandhawit et al., Geom. Topol. 22, No. 4, 2027–2114 (2018; Zbl 1395.57040)]. Their main result is a gluing theorem for the relative Bauer-Furuta invariants, which generalizes Manolescu’s construction [C. Manolescu, J. Differ. Geom. 76, No. 1, 117–153 (2007; Zbl 1128.57031)] by removing the assumption that the boundary of the manifold is a rational homology 3-sphere.
Manolescu’s construction of the relative Bauer-Furuta invariant is based on the Seiberg-Witten Floer spectrum, introduced for rational homology 3-spheres in [C. Manolescu, J. Differ. Geom. 76, No. 1, 117–153 (2007; Zbl 1128.57031)]. The more general unfolded version of the Seiberg-Witten Floer spectrum defined in the authors’ previous paper has two variations, the type-A “attractor” version and the type-R “repeller” version. The type-A version is an object in a category of ind-spectra, and the type-R version is an object in a category of pro-spectra.
Let \((Y_{\text{in}},\mathfrak{s}_{\text{in}})\) and \((Y_{\text{out}},\mathfrak{s}_{\text{out}})\) be two oriented \(\text{spin}^c\) 3-manifolds and \((X,\hat{\mathfrak{s}})\) a connected \(\text{spin}^c\) cobordism from \((Y_{\text{in}},\mathfrak{s}_{\text{in}})\) to \((Y_{\text{out}},\mathfrak{s}_{\text{out}})\), equipped with a Riemannian metric \(\hat{g}\) and a \(\text{spin}^c\) connection \(\hat{A}_0\). Denote the restrictions of \(\hat{g}\) and \(\hat{A}_0\) to \(Y_{\text{in}}\) by \(g_{\text{in}}\) and \(A_{\text{in}}\) and the restrictions of \(\hat{g}\) and \(\hat{A}_0\) to \(Y_{\text{out}}\) by \(g_{\text{out}}\) and \(A_{\text{out}}\).
The type-A unfolded relative Bauer-Furuta invariant of \(X\) is constructed as a morphism in a stable category \[ \begin{array}{cc} & \underline{\text{bf}}^A(X,\hat{\mathfrak{s}};S^1):\Sigma^{-(V_X^+\oplus V_{\text{in}}) }T(X,\hat{\mathfrak{s}};S^1)\wedge \underline{\text{swf}}^A(Y_{\text{in}},\mathfrak{s}_{\text{in}},A_{\text{in}},g_{\text{in}};S^1) \\ & \longrightarrow \underline{\text{swf}}^A(Y_{\text{out}},\mathfrak{s}_{\text{out}},A_{\text{out}},g_{\text{out}};S^1), \end{array} \] where \(\underline{\text{swf}}^A(Y_{\text{in (out)}},\mathfrak{s}_{\text{in (out)}},A_{\text{in (out})},g_{\text{in (out)}};S^1)\) are the type-A unfolded Seibert-Witten Floer spectra defined in the authors’ earlier paper, \(T(X,\hat{\mathfrak{s}};S^1)\) is the Thom spectrum of a virtual index bundle associated to a family of Dirac operators, \(V_X^+\) is a maximal positive subspace of \(\text{Im}(H^2(X,\partial X; \mathbb{R}) \rightarrow H^2(X;\mathbb{R}))\) with respect to the intersection form, and \(V_\text{in}\) is the cokernel of \(\iota^\ast: H^1(X; \mathbb{R}) \rightarrow H^1(Y_{\text{in}};\mathbb{R})\).
Similarly, the type-R unfolded relative Bauer-Furuta invariant of \(X\) is constructed as a morphism \[ \begin{array}{cc} & \underline{\text{bf}}^R(X,\hat{\mathfrak{s}};S^1):\Sigma^{-(V_X^+\oplus V_{\text{out}}) }T(X,\hat{\mathfrak{s}};S^1)\wedge \underline{\text{swf}}^R(Y_{\text{in}},\mathfrak{s}_{\text{in}},A_{\text{in}},g_{\text{in}};S^1) \\ & \longrightarrow \underline{\text{swf}}^R(Y_{\text{out}},\mathfrak{s}_{\text{out}},A_{\text{out}},g_{\text{out}};S^1), \end{array} \] where \(\underline{\text{swf}}^R(Y_{\text{in (out)}},\mathfrak{s}_{\text{in (out)}},A_{\text{in (out})},g_{\text{in (out)}};S^1)\) are the type-R unfolded Seibert-Witten Floer spectra defined in the authors’ earlier paper, and \(V_\text{out}\) is the cokernel of \(\iota^\ast: H^1(X; \mathbb{R}) \rightarrow H^1(Y_{\text{out}};\mathbb{R})\).
The first main theorem in the paper shows that both \( \underline{\text{bf}}^A(X,\hat{\mathfrak{s}};S^1)\) and \(\underline{\text{bf}}^R(X,\hat{\mathfrak{s}};S^1)\) are invariants of the 4-manifold \(X\).
Theorem 1.2. As one varies \((\hat{g}, \hat{A}_0)\), both domain and target of \(\underline{\text{bf}}^A(X,\hat{\mathfrak{s}};S^1)\) are changed by suspending or desuspending with the same number of copies of \(\mathbb{C}\); the morphism \(\underline{\text{bf}}^A(X,\hat{\mathfrak{s}};S^1)\) is invariant as a stable homotopy class. The same result holds for \(\underline{\text{bf}}^R(X,\hat{\mathfrak{s}};S^1)\). Moreover, when \(c_1(\mathfrak{s}|_Y)\) is torsion, one can construct further normalizations: \[ \begin{array}{cc} & \underline{\text{BF}}^A(X,\hat{\mathfrak{s}};S^1):\Sigma^{-(V_X^+\oplus V_{\text{in}}) }T(X,\hat{\mathfrak{s}};S^1)\wedge \underline{\text{SWF}}^A(Y_{\text{in}},\mathfrak{s}_{\text{in}};S^1) \\ & \longrightarrow \underline{\text{SWF}}^A(Y_{\text{out}},\mathfrak{s}_{\text{out}};S^1), \end{array} \]
\[ \begin{array}{cc} & \underline{\text{BF}}^R(X,\hat{\mathfrak{s}};S^1):\Sigma^{-(V_X^+\oplus V_{\text{out}}) }T(X,\hat{\mathfrak{s}};S^1)\wedge \underline{\text{SWF}}^R(Y_{\text{in}},\mathfrak{s}_{\text{in}};S^1) \\ & \longrightarrow \underline{\text{SWF}}^R(Y_{\text{out}},\mathfrak{s}_{\text{out}};S^1), \end{array} \] which are completely independent of metrics and base connections.
The second main theorem in the paper is a gluing theorem that works for general \(3\)-manifolds \(Y\). However, the authors utilize some additional homological assumptions in order to avoid working with more general categories and spectra.
Theorem 1.5. Let \[ (X_0,\hat{\mathfrak{s}}_0): (Y_0,{\mathfrak{s}}_0) \rightarrow (Y_2,{\mathfrak{s}}_2), \quad (X_1,\hat{\mathfrak{s}}_1): (Y_1,{\mathfrak{s}}_1) \rightarrow (-Y_2,{\mathfrak{s}}_2) \] be connected, \(\text{spin}^c\) cobordisms and \[ (X,\hat{\mathfrak{s}}): (Y_0,{\mathfrak{s}}_0) \sqcup (Y_1,\mathfrak{s}_1) \rightarrow \emptyset \] be the glued corbordism along \(Y_2\). If the following conditions hold

(i)

\(Y_2\) is connected,

(ii)

\(b_1(Y_0) = b_1(Y_1) = 0\),

(iii)

\(\text{im}(H^1(X_0;\mathbb{R}) \rightarrow H^1(Y_2;\mathbb{R})) \subset \text{im}(H^1(X_1;\mathbb{R}) \rightarrow H^1(Y_2;\mathbb{R}))\),

then, under the natural identification between domains and targets, one has \[ (1)\quad \quad BF(X,\hat{\mathfrak{s}})|_{\text{Pic}^0(X,Y_2)} = \tilde{\epsilon}(\underline{\text{bf}}^A(X_0,\hat{\mathfrak{s}}_0), \underline{\text{bf}}^R(X_1,\hat{\mathfrak{s}}_1)), \] where \(\tilde{\epsilon}(\cdot, \cdot)\) is the Spanier-Whitehead duality operation defined in Section 4.3 and the relative Picard torus \(\text{Pic}^0(X,Y_2)\) is given by \[ \text{ker}(H^1(X;\mathbb{R}) \rightarrow H^1(Y_2;\mathbb{R}))/ \text{ker}(H^1(X;\mathbb{Z}) \rightarrow H^1(Y_2;\mathbb{Z})). \]
In future work the authors intend to use their gluing theorem to compute the Bauer-Furuta invariants of irreducible 4-manifolds, drawing new results beyond those based on the Seiberg-Witten invariant.

Reviewer: David E. Hurtubise (Altoona)

MSC:

57K41	Invariants of 4-manifolds (including Donaldson and Seiberg-Witten invariants)
55P42	Stable homotopy theory, spectra
57R58	Floer homology

Keywords:

Seiberg-Witten-Floer spectra; Bauer-Furuta invariants; gluing; Spanier-Whitehead duality

Citations:

Zbl 1050.57024; Zbl 1395.57040; Zbl 1128.57031

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