  In this expository paper we present some ideas of algebraic topology in a
language accessible to non-specialists in the area. A $1$-cycle in a graph is a
set $C$ of edges such that every vertex is contained in an even number of edges
from $C$. It is easy to check that the sum (modulo $2$) of $1$-cycles is a
$1$-cycle. We start from the following problems: to find
  $\bullet$ the number of all $1$-cycles in a given graph;
  $\bullet$ a small number of $1$-cycles in a given graph such that any
$1$-cycle is the sum of some of them.
  We consider generalizations (of these problems) to graphs with symmetry, to
$2$-cycles in $2$-dimensional hypergraphs, and to certain configuration spaces
of graphs (namely, to the square and the deleted square).
