\(\underline{SW}\)=Milnor torsion
This paper announces the result that a version of the Seiberg-Witten invariant for certain smooth \(3\)-manifolds coincides with the Milnor torsion, as refined by Turaev. Let \(X\) be a compact oriented \(3\)-manifold such that \(b_1 (=\dim H^1 (X))>0\) and \(\partial X\) is a (possibly empty) union of tori, equipped with a homology orientation (if \(\partial X=\emptyset\) there is a canonical homology orientation given by Poincaré duality). Let \(\mathcal S\) denote the set of \(\operatorname{spin}^{\mathbb C}\)-structures on \(X\) such that \(\det s|\partial X\) is trivial. Consider pairs \((s,x)\) where \(s\in\mathcal S\) and \(x\in H=H^2 (X,\partial X )/\)torsion is a lift of the Chern class \(c_1 (\det s)\in H^2 (X)\). The Seiberg-Witten equations determined by \(s\) will have solutions \((A,\psi )\) where \(A\) is a connexion on \(\det s\). The curvature \(F_A\) determines an element of \(H\) which is a lift of \(c_1 (\det s)\). In the generic case the solution is an oriented \(0\)-dimensional manifold and \(SW(s,x)\) is defined to be the algebraic sum of all the solutions for which \(F_A =x\). If \(b_1 =1\) and \(\partial X=\emptyset\), then it is necessary to also choose a generator of \(H\). Now \(H^2 (X,\partial X)\) acts transitively on \(\mathcal S\) and so, if one chooses a `base' element \(y=(s_0 ,x_0 )\), one can define NEWLINE\[CARRIAGE_RETURNNEWLINESW_y (X)=\sum_{g\in H}\left(\sum_{h\equiv g}SW(h\cdot s_0,x_0 +2g)\right)g\in \mathbb ZH CARRIAGE_RETURNNEWLINE\]NEWLINE where \(h\) ranges over the (finite) set of lifts of \(g\) into \(H^2 (X,\partial X)\) and \(\mathbb Z H\) is the set of all \(\mathbb Z\)-valued functions on \(H\). Then, to remove the dependency on \(y\), define \(\underline{SW}(X)\) to be the orbit of \(SW_y (X)\) in \(\mathbb Z H/H\). In fact if \(b_1 >1\), then \(SW_y (X)\in\mathbb Z [H]\), where \(\mathbb Z [H]\) is the ring of finitely-supported \(\mathbb Z \)-valued functions on \(H\), i.e. the integral group ring of \(H\). If \(b_1 =1\), then \((1-t)^{\epsilon (X)}SW_y (X)\in\mathbb Z [H]\), where NEWLINE\[CARRIAGE_RETURNNEWLINE\varepsilon (X)=\begin{cases} 2 & \text{ if } \partial X=\emptyset \\ 1 & \text{otherwise}\end{cases}.CARRIAGE_RETURNNEWLINE\]NEWLINE Thus, in any case, \(\underline{SW}\in\mathbf Q (H)/H\), where \(\mathbf Q (H)\) is the quotient field of \(\mathbb Z [H]\). The classical Reidemeister torsion \(\tau\) of a polyhedron \(X\) is defined from a multiplicative map NEWLINE\[CARRIAGE_RETURNNEWLINE\begin{tikzcd}[column sep=huge] \pi =\pi_1 (X)\ar[r,"h"] & F \end{tikzcd}CARRIAGE_RETURNNEWLINE\]NEWLINE where \(F\) is a field such that the chain complex \(C_{\ast}(X)\otimes_h F\) is acyclic (\(\tau=0\) if it is not acyclic). When \(X\) is an odd-dimensional manifold and \(F\) is the quotient field of \(H=H_1 (X)/\)torsion, this is called the Milnor torsion. Classically \(\tau\in F/\pm h(\pi )\), but Turaev has shown how to use a homology orientation of \(X\) to remove the sign ambiguity. Thus \(\tau\in\mathbf Q (H)/H\). The main result of this announcement is that \(\underline{SW} =\tau\) when \(X\) is as assumed in the first paragraph. An outline of the proof is presented. In the case when \(X\) is the complement of a link in \(S^3\), then \(\tau\) is closely related to the multi-variable Alexander polynomial \(\Delta\) of the link. The proof in this case uses the axiomatic skein-theory of Conway and Turaev for \(\Delta\). If \(X\) is the complement of a link in a rational homology sphere, then non-zero Dehn surgeries convert \(X\) into a link complement in \(S^3\). The fact that \(\underline{SW}\) and \(\tau\) satisfy identical surgery formulae completes the proof in this case. In general \(X\) can be converted into a link complement in a rational homology sphere by removing some solid tori. The proof now follows from the fact that \(\underline{SW}\) and \(\tau\) satisfy identical gluing formulae. Recently a possible sharpening of this theorem has been conjectured. Turaev has defined the notion of an Euler structure on a cell complex \(X\) with Euler characteristic \(0\). The set of Euler structures is acted on freely and transitively by \(H_1 (X)\). An Euler structure \(\theta\) (together with a homology orientation) determines a torsion \(\tau_{\theta}\in\mathbb Z H_1 (X)\) with no ambiguity. Now a \(\operatorname{Spin}^{\mathbb C}\)-structure \(\theta\) on a \(3\)-manifold \(X\) determines, by an obvious refinement of the definition of \(SW_y\) above, an element \(SW_{\theta}\in\mathbb Z H_1 (X)\). Turaev has shown that, for a closed \(3\)-manifold \(X\), there is a 1:1 correspondence between the set of \(\operatorname{Spin}^{\mathbb C}\)-structures and the set of Euler structures on \(X\). The conjecture is that \(\tau_{\theta}=SW_{\theta}\).
scientific article; zbMATH DE number 980880