The \(\frac {10}{8}\)-conjecture and equivariant \(e_C\)-invariants (Q1890249)

The paper provides a proof of the so-called 10/8-conjecture: Let \(X\) be a smooth closed oriented non-spin 4-manifold with even intersection form. Thus the intersection form is indefinite by the Donaldson theorem and therefore isomorphic to \(kE_8\oplus nH\), \(n\geq 1\), by standard classification of indefinite forms (\(E_8\) is the even definite form of rank 8 and index 8 associated to the Lie algebra \(E_8\), and \(H\) is the hyperbolic plane). Then the 10/8-conjecture asserts that \(n > |k|\) i.e. the ratio of the second Betti number of \(X\) (= rank of the intersection form) to the absolute value of the signature of \(X\) is greater than or equal to 10/8. The author proves also the following estimate for a covering of \(X\): First recall that by a recent result of R. Lee and T. J. Li any manifold \(X\) as described above admits a \(2^p\)-fold Spin covering manifold \(M\). Let \(2k_1E_8\oplus n_1H\) be the intersection form of \(M\). Then \(n_1\geq 2|k_1|+ 2^p- 1+ b_1(M)- 2^p b_1(X)\).