An analytic computation of \(ko_{4\nu -1}(BQ_ 8)\) (Q1918511)

The connectivity \(K\)-theory groups \(ko_*(B \pi)\) of a group \(\pi\) appear in many contexts; for example, they are the building blocks for equivariant spin bordism at the prime 2. They also play an important role in the Gromov-Lawson-Rosenberg conjecture which was the starting point of the second author's original investigation [IHES preprint IHES-M-94-62]. He first studied the eta invariant, which is an analytic invariant. In this paper, the authors use the eta invariant to determine the additive structure of \(ko_{4\nu - 1} (BQ_8)\), where \(Q_8 = \{\pm 1, \pm i, \pm j, \pm k\}\) is the quaternion group of order 8. The main results are the following theorems: Theorem 1. \[ ko_{8 \mu + 3} (BQ_8) \cong (\mathbb{Z}/2^{3 + 4 \mu}) \oplus (\mathbb{Z}/2^{2 \mu}) \oplus (\mathbb{Z}/2^{2 \mu + 2}) \oplus (\mathbb{Z}/2^{2 \mu + 2}). \tag{a} \] \[ ko_{8 \mu + 7} (BQ_8) \cong (\mathbb{Z}/2^{6+4 \mu}) \oplus (\mathbb{Z}/2^{2 \mu}) \oplus (\mathbb{Z}/2^{2 \mu + 2}) \oplus (\mathbb{Z}/2^{2 \mu + 2}). \tag{b} \] Theorem 3. Let \(R_0 (\pi)\) be the augmentation ideal of all virtual representations of virtual dimension 0 in the group representation ring \(R (\pi)\). Let \((M,g,s, \sigma)\) denote a closed manifold of dimension \(m\) with a Riemannian metric \(g\), a spin structure \(s\), and a \(\pi\) structure \(\sigma\). If \(m\) is odd, let \(D_\rho\) be the Dirac operator on \(M\) with coefficients in the flat bundle determined by a representation \(\rho\) of \(\pi\). Define \(\eta (M) (\rho) = \eta (M,g,s, \sigma) (\rho) : = \eta (D_\rho ) \in \mathbb{R}\). Let \(\rho \in R_0 (\pi)\) and \(m\) be odd. Then the homomorphism \(\eta(\rho) : M \text{Spin}_m (B \pi) \to \mathbb{R}/ \mathbb{Z}\) which maps a class represented by \((M,s, \sigma)\) in dimension \(m\) to \(\eta (M,g,s, \sigma)\) \((\rho)\) is well defined. Furthermore, if \(\rho\) is of real type and \(m \equiv 3 \pmod 8\) or if \(\rho\) is of quaternion type and \(m \equiv 7 \pmod 8\), the range of \(\eta (\rho)\) can be recplaced by \(\mathbb{R}/2 \mathbb{Z}\). The authors use the following theorem to extend the eta invariant to a map in \(K\)-theory. Theorem 4. Let \(\pi\) be a finite group, let \(\rho \in R_0 (\pi)\), and let \(m\) be odd. Then the homomorphism \(\eta^{ko} (\rho) : (ko_m (B\pi))_{(2)} \to (\mathbb{R}/ \mathbb{Z})_{(2)}\) which maps a class represented by \((M,s, \sigma)\) in dimension \(m\) to \(\eta (M,g,s, \sigma) (\rho)\) is well defined when localized at the prime 2. Furthermore, if \(\rho\) is of real type and \(m \equiv 3 \pmod 8\) or if \(\rho\) is of quaternion type and \(m \equiv 7 \pmod 8\), the range of \(\eta^{ko}(\rho)\) can be replaced by \((\mathbb{R}/2 \mathbb{Z})_{(2)}\). A last result refers to the spherical space forms for which the authors use a generalized Atiyah-Patodi-Singer theory given by \textit{H. Donnelly} [Indiana Univ. Math. J. 27, 889-918 (1978; Zbl 0402.58006)].