The \(\eta\)-invariant and Pontryagin duality in \(K\)-theory.

*(English. Russian original)*Zbl 1030.58017
Math. Notes 71, No. 2, 245-261 (2002); translation from Mat. Zametki 71, No. 2, 271-291 (2002).

The paper concerns relations between some spectral invariants of (a generalization of) elliptic operators on closed manifolds and topological invariants, and the more traditional part of the results of the paper is an expression of the fractional part of the Atiyah-Patodi-Singer eta-invariant in terms of the so-called linking number in \(K\)-theory (with coefficients) and a construction of an elliptic even-order operator on an odd-dimensional manifold with nontrivial fractional part of the eta-invariant (suitable example, due to Gilkey, in the case of even-dimensional is the \(\text{Pin}^c\)-operator on the real projective space \(\mathbb{R} P^{2n}\)).

In order to give a more detailed presentation of the results of the paper let us recall some earlier results of the authors. Namely, in some earlier papers, the authors developed a theory of elliptic operators on so-called pseudodifferential subspaces of \(C^\infty(M, E)\) (smooth sections of a vector bundle \(E\) over a smooth closed manifold \(M\)), i.e. subspaces that are images of pseudodifferential projections. A symbol of such a pseudodifferential subspace is well defined as a bundle on the cosphere tangent bundle, and parity (odd, even) of such subspace has been defined in terms of its symbol. Moreover a homotopy-invariant “dimension” functional \(d\) is well defined on such pseudodifferential subspaces in the case the parities of the subspace and dimension of the manifold in question are (even, odd) or (odd, even). For example the subspace corresponding to the nonnegative eigenvalues of an elliptic self-adjoint operator \(A\) of nonnegative order is a pseudodifferential subspace, and its “dimension” \(d\) equals the eta-invariant of the operator \(A\) (suitable parities are assumed).

For elliptic operators \(D\) acting between pseudodifferential subspaces \(\widehat L^1\), \(\widehat L^2\) the authors also proved (in an earlier paper) an index theorem: \(\text{index}(D, \widetilde L^1, \widetilde L^2)=\tfrac 12\text{ index }\widetilde D+ d(\widetilde L^1)- d(\widetilde L^2)\), where \(\widetilde D\) is an auxiliary elliptic operator built out of the operator \(D\). It follows from this theorem that the symbol of a pseudodifferential subspace determines a two-torsion element of \((K(S^\bullet M)/K(M))\). Moreover it has been proved that the group of stable homotopy classes of elliptic pseudodifferential operators acting between pseudodifferential subspaces is isomorphic to the \(K\)-theory with coefficients in \(\mathbb{Q}/\mathbb{Z}\) of the cotangent bundle \(T^\bullet M\).

The main results of the present paper is a formula, which expresses the fractional part of \(2d(\widehat L)\) as the \(K\)-theoretical linking number of the pseudodifferential subspace in question and the orientation bundle of the manifold \(M\). This can be summarized as follows. Using the index theorem above the authors express the fractional part of \(2d(\widehat L)\) as the index \(\text{mod }2^N\) of certain elliptic operator acting between pseudodifferential subspaces. Next the authors introduce suitable Pontryagin duality in \(K\)-theory with coefficients, which provides identification \(K^i_c(T^\bullet M,\mathbb{Q}/\mathbb{Z})\approx\operatorname{Hom}(K^i(M),\mathbb{Q}/\mathbb{Z})\), and using this identification define, in a purely topological manner, a linking form \(\bigcap: \text{Tor }K^{i-1}_c(T^\bullet M)\times \text{Tor }K^i(M)\to\mathbb{Q}/\mathbb{Z}\). The form is proved to be nondegenerate, and using the above-mentioned identification of stable homotopy classes of elliptic pseudodifferential operators acting between pseudodifferential subspaces and the \(K\)-theory with coefficients in \(\mathbb{Q}/\mathbb{Z}\) of the cotangent bundle \(T^\bullet M\) the authors show that the linking form can also be expressed as the index \(\text{mod }2^N\) of an elliptic operator. Finally, the operator that appears in the above-mentioned formula for the fractional part of \(2d(\widehat L)\) is identified as an operator in the “index” formula for the linking pairing, and therefore the fractional part of \(2d(\widehat L)\) is expressed as the \(K\)-theoretical linking number of the pseudodifferential subspace in question and the orientation bundle of the manifold \(M\), as desired.

In order to give a more detailed presentation of the results of the paper let us recall some earlier results of the authors. Namely, in some earlier papers, the authors developed a theory of elliptic operators on so-called pseudodifferential subspaces of \(C^\infty(M, E)\) (smooth sections of a vector bundle \(E\) over a smooth closed manifold \(M\)), i.e. subspaces that are images of pseudodifferential projections. A symbol of such a pseudodifferential subspace is well defined as a bundle on the cosphere tangent bundle, and parity (odd, even) of such subspace has been defined in terms of its symbol. Moreover a homotopy-invariant “dimension” functional \(d\) is well defined on such pseudodifferential subspaces in the case the parities of the subspace and dimension of the manifold in question are (even, odd) or (odd, even). For example the subspace corresponding to the nonnegative eigenvalues of an elliptic self-adjoint operator \(A\) of nonnegative order is a pseudodifferential subspace, and its “dimension” \(d\) equals the eta-invariant of the operator \(A\) (suitable parities are assumed).

For elliptic operators \(D\) acting between pseudodifferential subspaces \(\widehat L^1\), \(\widehat L^2\) the authors also proved (in an earlier paper) an index theorem: \(\text{index}(D, \widetilde L^1, \widetilde L^2)=\tfrac 12\text{ index }\widetilde D+ d(\widetilde L^1)- d(\widetilde L^2)\), where \(\widetilde D\) is an auxiliary elliptic operator built out of the operator \(D\). It follows from this theorem that the symbol of a pseudodifferential subspace determines a two-torsion element of \((K(S^\bullet M)/K(M))\). Moreover it has been proved that the group of stable homotopy classes of elliptic pseudodifferential operators acting between pseudodifferential subspaces is isomorphic to the \(K\)-theory with coefficients in \(\mathbb{Q}/\mathbb{Z}\) of the cotangent bundle \(T^\bullet M\).

The main results of the present paper is a formula, which expresses the fractional part of \(2d(\widehat L)\) as the \(K\)-theoretical linking number of the pseudodifferential subspace in question and the orientation bundle of the manifold \(M\). This can be summarized as follows. Using the index theorem above the authors express the fractional part of \(2d(\widehat L)\) as the index \(\text{mod }2^N\) of certain elliptic operator acting between pseudodifferential subspaces. Next the authors introduce suitable Pontryagin duality in \(K\)-theory with coefficients, which provides identification \(K^i_c(T^\bullet M,\mathbb{Q}/\mathbb{Z})\approx\operatorname{Hom}(K^i(M),\mathbb{Q}/\mathbb{Z})\), and using this identification define, in a purely topological manner, a linking form \(\bigcap: \text{Tor }K^{i-1}_c(T^\bullet M)\times \text{Tor }K^i(M)\to\mathbb{Q}/\mathbb{Z}\). The form is proved to be nondegenerate, and using the above-mentioned identification of stable homotopy classes of elliptic pseudodifferential operators acting between pseudodifferential subspaces and the \(K\)-theory with coefficients in \(\mathbb{Q}/\mathbb{Z}\) of the cotangent bundle \(T^\bullet M\) the authors show that the linking form can also be expressed as the index \(\text{mod }2^N\) of an elliptic operator. Finally, the operator that appears in the above-mentioned formula for the fractional part of \(2d(\widehat L)\) is identified as an operator in the “index” formula for the linking pairing, and therefore the fractional part of \(2d(\widehat L)\) is expressed as the \(K\)-theoretical linking number of the pseudodifferential subspace in question and the orientation bundle of the manifold \(M\), as desired.

Reviewer: Wieslaw Oledzki (Bialystok)

##### MSC:

58J28 | Eta-invariants, Chern-Simons invariants |

58J20 | Index theory and related fixed-point theorems on manifolds |

58J22 | Exotic index theories on manifolds |

55N15 | Topological \(K\)-theory |