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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].

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