A coboundary expander is a simplicial complex in which the norm of a cochain's coboundary controls
its distance from the space of coboundaries. For a
finite pure 
-dimensional complex 
, let 
denote its faces with 
vertices and use cochains with coefficients in the two-element
field. Assign each 
-face 
the weight
 |
The norm 
is the sum of the weights of the faces where 
is nonzero. The coboundary
operator 
sums 
over the codimension-one faces of an 
-face, modulo 2. Write 
, using constant cochains for the reduced degree-zero
space 
.
The complex is an 
-coboundary
expander, for 
,
when the expansion inequality
 |
holds for every cochain 
and every degree 
. For degree zero this compares an edge boundary
with the smaller weighted side of a vertex partition.
The inequality forces vanishing of reduced cohomology in these degrees, since every cocycle must have zero distance from the space of coboundaries.
