  For a fixed 2-block Springer fiber, we describe the structure of its
irreducible components and their relation to the Bialynicki-Birula paving,
following work of Fung. That is, we consider the space of complete flags in C^n
preserved by a fixed nilpotent matrix with 2 Jordan blocks, and study the
action of diagonal matrices commuting with our fixed nilpotent. In particular,
we describe the structure of each component, its set of torus fixed points, and
prove a conjecture of Fung describing the intersection of any pair.
  Then we define a convolution algebra structure on the direct sum of the
cohomologies of pairwise intersections of irreducible components and closures
of C^*-attracting sets (that is, Bialynicki-Birula cells), and show this is
isomorphic to a generalization of the arc algebra of Khovanov defined by the
first author. We investigate the connection of this algebra to Cautis &
Kamnitzer's recent work on link homology via coherent sheaves and suggest
directions for future research.
