  Graphs with given k vertices generate an (acyclic) simplicial complex. We
describe the homology of its quotient complex, formed by all connected graphs,
and demonstrate its applications to the topology of braid groups, knot theory,
combinatorics, and singularity theory. The multidimensional analogues of this
complex are indicated, which arise naturally in the homotopy theory, higher
Chern-Simons theory and complexity theory.
