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].

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].

Reviewer: Akira Asada (Takarazuka)