We present new methods to obtain singular twisted sums
X
⊕
Ω
X
X\oplus _\Omega X
(i.e., exact sequences
0
→
X
→
X
⊕
Ω
X
→
X
→
0
0\to X\to X\oplus _\Omega X \to X\to 0
in which the quotient map is strictly singular) when
X
X
is an interpolation space arising from a complex interpolation scheme and
Ω
\Omega
is the induced centralizer.
Although our methods are quite general, we are mainly concerned with the choice of
X
X
as either a Hilbert space or Ferenczi’s uniformly convex Hereditarily Indecomposable space. In the first case, we construct new singular twisted Hilbert spaces (which includes the only known example so far: the Kalton-Peck space
Z
2
Z_2
). In the second case we obtain the first example of an H.I. twisted sum of an H.I. space.
During our study of singularity we introduce the notion of a disjointly singular twisted sum of Köthe function spaces and construct several examples involving reflexive
p
p
-convex Köthe function spaces (which includes the function space version of the Kalton-Peck space
Z
2
Z_2
).
We then use Rochberg’s description of iterated twisted sums to show that there is a sequence
F
n
\mathcal F_n
of H.I. spaces so that
F
m
+
n
\mathcal F_{m+n}
is a singular twisted sum of
F
m
\mathcal F_m
and
F
n
\mathcal F_n
, while for
l
>
n
l>n
the direct sum
F
n
⊕
F
l
+
m
\mathcal F_n \oplus \mathcal F_{l+m}
is a nontrivial twisted sum of
F
l
\mathcal F_l
and
F
m
+
n
\mathcal F_{m+n}
.