  We show that superconformal ${\cal N}=4,2$ algebras are well-suited to
represent some invariant constructions characterizing exotic $\mathbb{R}^4$
relative to a given radial family. We examine the case of ${\cal N}=4,
\hat{c}=4$ (at $r=1$ level) superconformal algebra which is realized on flat
$\mathbb{R}^4$ and curved $S^3\times \mathbb{R}$. While the first realization
corresponds naturally to standard smooth $\mathbb{R}^4$ the second describes
the algebraic end of some small exotic smooth $\mathbb{R}^4$'s from the radial
family of DeMichelis-Freedman and represents the linear dilaton background
$SU(2)_k\times \mathbb{R}_Q$ of superstring theory.
  From the modular properties of the characters of the algebras one derives
Witten-Reshetikhin-Turaev and Chern-Simons invariants of homology 3-spheres.
These invariants are represented rather by false, quasi-modular, Ramanujan
mock-type functions. Given the homology 3-spheres one determines exotic smooth
structures of Freedman on $S^3\times \mathbb{R}$. In this way the fake ends are
related to the SCA ${\cal N}=4$ characters.
  The case of the ends of small exotic $\mathbb{R}^4$'s is more complicated.
One estimates the complexity of exotic $\mathbb{R}^4$ by the minimal complexity
of some separating from the infinity 3-dimensional submanifold. These
separating manifolds can be chosen, in some exotic $\mathbb{R}^4$'s, to be
homology 3-spheres. The invariants of such homology 3-spheres are, again,
obtained from the characters of SCA, ${\cal N}=4$.
  Next we take into account the modification of the algebra of modular forms
due to the noncommutativity of the codimension-one foliations of the homology
3-spheres. Then, the modification of modular forms is represented by the
Connes-Moscovici construction ...
