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The $$\eta$$ invariant and elliptic operators in subspaces. (English) Zbl 1029.58016
de Gosson, Maurice (ed.), Jean Leray ’99 conference proceedings. The Karlskrona conference, Sweden, August 1999 in honor of Jean Leray. Dordrecht: Kluwer Academic Publishers. Math. Phys. Stud. 24, 373-387 (2003).
This is a report of the authors’ study on fractional parts of $$\eta$$-invariants [A. Yu. Savin, B. W. Schulze and B. Sternin, $$K$$-Theory 27, 253-272 (2002; Zbl 1021.58017); A. Yu. Savin, B. Sternin, Sb. Math. 190, 1195-1228 (1999; Zbl 0963.58008) and ibid. 191, 1191-1213 (2000; Zbl 0981.58018)].
The $$\eta$$-invariant of an operator $$A$$ satisfying the parity condition $\text{ord }A+ \dim M\equiv 1\pmod 2,\tag{1}$ is rigid [P. Gilkey, Lect. Notes Math. 1410, 202-211 (1989; Zbl 0820.58054)]. The authors show that the $$\eta$$-invariant of $$A$$ is completely determined by the nonnegative spectral subspace $$\widehat L_+(A)$$ of $$A$$ if $$A$$ satisfies (1). Then applying the index theorem of subspaces, the computation of the fractional part of the $$\eta$$-invariant is reduced to the ‘index modulo $$n$$’ problem for operators in subspaces, which is computed by the direct image of $$K$$-theory with coefficients in $$\mathbb{Z}_n$$. As an application, it is shown that the $$\eta$$-invariant of an elliptic operator on orientable manifolds satisfying (1) has at most 2 in its denominator [cf. P. Gilkey, Adv. Math. 58, 243-284 (1985; Zbl 0602.58041)].
The outline of the paper is as follows: In Sect. 2, after explaining the $$\eta$$-invariant, the pseudo-differential subspace $$\widehat L\subset C^\infty(M, E)$$ is defined to be the image of an order zero pseudo-differential projection $$P: C^\infty(M, E)\to C^\infty(M, E)$$, together with its parity. An operator in subspaces is the restriction of a pseudo-differential operator $$D: D\widehat L_1\subset\widehat L_2$$ on $$\widehat L_1$$. If $$\widehat L_1$$ and $$\widehat L_2$$ have same parity, a classical elliptic operator $$\widetilde D$$ is defined and if the parity condition (1) is satisfied, it is shown $\text{ind}(D: \widehat L_1\to\widehat L_2)= \textstyle{{1\over 2}}\text{ ind }\widetilde D+ d(\widehat L_1)- d(\widehat L_2)$ (Sect. 3, Th. 3.2). Applying this theorem to $$d+\delta$$ acting on the nonnegative spectral subspace, it is derived $$\{\eta(d+ \delta)\}= \{\chi(M)/2\}$$ (Sect. 3). To extend this result, elliptic operators modulo $$n$$ and the group of their stable homotopy classes $$\text{Ell}(M,\mathbb{Z}_n)$$ are defined in Sect. 4. There is an isomorphism of groups $$\chi_n: \text{Ell}(M, \mathbb{Z}_n)\cong K_c(T^* M,\mathbb{Z}_n)$$, and it is shown that the modulo $$n$$-index is the composition of $$\chi_n$$ and the direct image map $$p!: K(T^* M,\mathbb{Z}_n)\to \mathbb{Z}_n$$ (Th. 4.2). By using this result, a formula for the fractional part of the $$\eta$$-invariant is given in Sect. 5 (Sect. 5.1 and 5.2) which provides the first mentioned result on the denominator of the $$\eta$$-invariant.
For the entire collection see [Zbl 1017.00056].

##### MSC:
 58J28 Eta-invariants, Chern-Simons invariants 14L99 Algebraic groups