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\(\eta\)-invariants and Rokhlin congruences. (English. Abridged French version) Zbl 0767.57015
By a classical theorem Atiyah-Hirzebruch, a spin manifold \(M\) of dimension \(8k + 4\) has \(\hat A(M) \equiv 0(2)\). The author sketches the proof of a generalization of this formula to the case where \(M\) is only assumed to be oriented; \(\hat A(M)\) is then congruent \(\mod 2\mathbb{Z}\) to a certain number defined by means of a submanifold \(B^{8k + 2}\) of \(M\) dual to \(w_ 2(M)\). In the case of 4-dimensional manifolds, this generalization is due to Rohklin.
The proof is completely analytic: \(\hat A(M)\) is interpreted as an \(\eta\)-invariant by means of the Atiyah-Patodi-Singer index theorem. This \(\eta\)-invariant is then computed by passing to the adiabatic limit following Bismus-Cheeger.

57R20 Characteristic classes and numbers in differential topology
58J20 Index theory and related fixed-point theorems on manifolds