A twisted sum of two Banach spaces \(Y\) and \(Z\) is a Banach space \(X\) containing an isomorphic copy of \(Y\) such that the respective quotient space is isomorphic to \(Z\). Using homological language, \(X\) is a twisted sum of \(Y\) and \(Z\) if there exists a short exact sequence \[ 0 \longrightarrow Y \longrightarrow X \longrightarrow Z \longrightarrow 0.\] The twisted sum \(X\) is said to be trivial when the copy of \(Y\) is complemented on \(X\). The exact sequences for which the quotient map \(X \longrightarrow Z\) is strictly singular are called singular twisted sums. This notion is of major interest in the theory and corresponds in some sense to the opposite of triviality.
In the paper under review, the complex interpolation method jointly with a fragmentation-amalgamation technique is used to construct new examples of non-trivial, singular, and strictly non-singular twisted sums.
The authors study complex interpolation for some couples of unconditional sums of Banach spaces and compute their respective derivations. These results are applied to unconditional sums of finite dimensional spaces. Among the main results, we highlight:

1.

Construction of a non-trivial \(\ell_2\)-fragmentation of the Kalton-Peck space with the property of strict non-singularity. This space is also related with the Enflo-Lindenstrauss-Pisier space which is the first example of non-trivial twisted Hilbert space.

2.

Construction of non-trivial twisted Hilbert spaces including an asymptotically hilbertian space.

3.

Singularity of the derivation of some Lorentz sequence spaces. In particular, they prove that the conditions for singularity for twisted sums of sequence spaces obtained by derivations proved recently by [J. M. F. Castillo et al., Trans. Am. Math. Soc. 369, No. 7, 4671–4708 (2017; Zbl 1427.46050)] are not necessary.