Kreck-Stolz invariants for quaternionic line bundles (Q2838120)

The authors introduce a new invariant for certain quaternionic line bundles, called the \(t\)-invariant, which is a quaternionic version of the Kreck-Stolz invariants of certain complex line bundles [\textit{M. Kreck} and \textit{S. Stolz}, Ann. Math. (2) 127, No. 2, 373--388 (1988; Zbl 0649.53029)].NEWLINENEWLINEIn this paper, the set of isomorphism classes of quaternionic line bundles over a manifold \(M\) is denoted by \(\text{Bun}(M)\). The second Chern class defines a function \(c_2:\text{Bun}(M)\to H^4(M)\), and \(\text{Bun}_0(M)\subset \text{Bun}(M)\) is the subset defined by the bundles for which this class is torsion. Suppose that \(M\) is closed, spin, of dimension \(4k-1\), and with \(H^3(M;\mathbb{Q})=0\). Then the \(t\)-invariant is defined as a function \(t_M:\text{Bun}_0(M)\to\mathbb{Q}/\mathbb{Z}\).NEWLINENEWLINEFor \(E\in\text{Bun}_0(M)\), a simpler first definition of \(t_M(E)\) is given under the extra condition that \(M\) bounds a compact spin manifold \(W\) so that \(E\) is the restriction of a quaternionic line bundle \(\overline E\) over \(W\). On the one hand, there is a characteristic class \(\text{ch}'(\overline{E})\) such that \(2-\text{ch}(\overline{E})=c_2(\overline{E})\,\text{ch}'(\overline{E})\). On the other hand, since \(c_2(E)=0\in H^4(M;\mathbb{Q})\), the class \(c_2(\overline{E})\in H^4(W;\mathbb{Q})\) has a unique lift \(\bar c_2(\overline{E})\in H^4(W,M;\mathbb{Q})\). Then \(t_M(E)\in\mathbb{Q}/\mathbb{Z}\) is given by \(-\frac{1}{a_{k+1}}\) times the evaluation at \([W,M]\) of \(\widehat{A}(TW)\,\text{ch}'(\overline{E})\,\bar c_2(\overline{E})\), where \(a_j\) is \(1\) if \(j\) is even and \(2\) if \(j\) is odd. The Atiyah-Singer index theorem for twisted Dirac operators on closed spin \(4k\)-manifolds is used to show that this definition of \(t_M(E)\) is independent of the choice of \(W\) and \(\overline E\). Then the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary is used to extend this definition of \(t_M(E)\), removing the assumption on the existence of \(W\) and \(\overline{E}\).NEWLINENEWLINEThe rest of the paper contains a deep study of this \(t\)-invariant and its applications. First, it is proved that the \(t\)-invariant vanishes on trivial bundles, is additive under connected sums, and natural under almost-diffeomorphisms. In the case of homology spheres \(\Sigma^{4k-1}\) with \(k\geq3\), \(t_\Sigma\) is expressed in terms of the Adams \(e\)-invariant. The authors combine \(t_M\) with the Eells-Kuiper invariant \(\mu(M)\) to classify the diffeomorphism types of \(2\)-connected rational homology \(7\)-spheres \(M\). It is shown that an orientation preserving homeomorphism between closed smooth oriented \(2\)-connected rational homology spheres is exotic if and only if it does not preserve the \(t\)-invariants, where exotic means that it is not homotopic to a piecewise linear homeomorphism. Finally, let \(k\geq2\) and \(c\in\mathbb{Z}\), and let \(H\) be the tautological quaternionic line bundle over \(\mathbb{H}P^{k-1}\). Then a condition involving the \(t\)-invariant is given for the existence of some quaternionic line bundle \(E_c\) over \(\mathbb{H}P^{k-1}\) so that \(c_2(E_c)=c\,c_2(H)\). Moreover computations are made in many examples, and it is conjectured that simply connected spin \(7\)-manifolds \(M\) with \(H^3(M)=0\) are classified by their Eells-Kuiper invariant, their Massey product structure, and their \(t\)-invariants.