Cycles in graphs and in hypergraphs: towards homology theory

In this expository paper we present some ideas of algebraic topology (more precisely, of homology theory) 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).