On the vanishing of the Rokhlin invariant
It was proven by Casson in the 1980's that the Rokhlin invariant \(\mu(M)\in \mathbb{Z}/16 \mathbb{Z}\) of an amphichiral homology \(3\)-sphere \(M\) vanishes, (see e.g. [\textit{S. Akbulut} and \textit{J. D. McCarthy}, Casson's invariant for oriented homology 3-spheres. An exposition. Mathematical Notes, 36. Princeton, NJ: Princeton University Press. (1990; Zbl 0695.57011)]). In the article under review, a new proof of this theorem is presented.NEWLINENEWLINEA manifold \(M\) is called amphichiral, if it admits an orientation-reversing self-homeomorphism. The Rokhlin invariant of a closed oriented spin 3-manifold is defined as \(\mu(M)=\text{Sign\;} X\;(\text{mod\;}16)\), where \(X\) is a compact spin 4-manifold with boundary \(M\). If \(H^1(M,\mathbb{Z}/2\mathbb{Z})=0\), then the spin structure on \(M\) is unique, and thus the Rokhlin invariant depends only on the diffeomorphism type of \(M\).NEWLINENEWLINEThe new proof uses \(e\)-manifolds which were already introduced and studied in [\textit{T. Moriyama}, J. Math. Sci., Tokyo 18, No. 2, 193--237 (2011; Zbl 1259.57008)]. An \(n\)-dimensional \(e\)-manifold is a triple \((W,V,e)\) where \(W\) is an \(n\)-dimensional compact manifold, possibly with boundary, and where \(V\) is a submanifold of codimension \(3\), such that \(\partial V=V\cap \partial W\) with transversal intersection. Furthermore, \(e\) is a class in \(H^2(W\setminus V; \mathbb{Q})\) such that the restriction of \(e\) to a sphere in the normal bundle of \(V\) in \(W\) is the Euler class of this sphere. The relevant \(e\)-manifolds in the article carry spin structures, i.e.\ a spin structure on \(V\) and \(W\). A rational-valued invariant \(\sigma(\alpha)\) was defined in [loc. cit.] for 6-dimensional \(e\)-manifolds \(\alpha\). If \(\alpha\) is the boundary of the \(7\)-dimensional \(e\)-manifold \(\beta=(Z,X,e)\), then \(\sigma(\partial \beta)= \text{Sign\;} X\).NEWLINENEWLINEThe author associates to every homology \(3\)-sphere \(M\) a \(6\)-dimensional spin \(e\)-manifold \(\alpha_M\). He shows that \(\alpha_M\) is the boundary of a \(7\)-dimensional \(e\)-manifold, and thus \(\mu(M)= \sigma(\alpha_M) (\text{mod\;}16)\). On the other hand \(\sigma(\alpha_{-M})= - \sigma(\alpha_M)\), and hence \(\sigma(\alpha_M)\) and \(\mu(M)\) vanish for amphichiral homology spheres.NEWLINENEWLINEThe author also defines bordims classes \(\Omega_n^{e,\text{spin}}\) of \(n\)-dimensional closed spin \(e\)-manifolds, and he shows that there is a unique isomorphism \(\Phi:\Omega_6^{e,\text{spin}}\to \mathbb{Q}/16\mathbb{Z}\oplus \mathbb{Q}/4\mathbb{Z}\) such that NEWLINE\[NEWLINE\Phi([W,\emptyset,e])=\left(\frac16\int_W p_1(TW)e-e^3,\frac12 \int_W e^3\right).NEWLINE\]NEWLINE This bordism class nicely fits into an exact sequence NEWLINE\[NEWLINE0\to \Omega_4^{\text{spin}}(BSpin(3))\to \Omega_6^{\text{spin}}(K(\mathbb{Q},2))\to \Omega_6^{e,\text{spin}}\to 0NEWLINE\]NEWLINE which is studied further.NEWLINENEWLINEThe article is very well-written and can be read fluently. The new proof of the vanishing of the Rokhlin invariant is according to the author more direct than previous proofs.
scientific article; zbMATH DE number 6068677