Intuitive explanation of singular homology Let $J$ be a $\mathcal{C}^1$-functional over a inner product space. The local behavior of $J$ near an isolated critical point $u$ is described by the sequence of critical groups
$$C_q(J, u):= H_q(J^c\cap U, J^c\cap U\setminus\{0\}), \qquad q\ge 0,$$
where $c=J(u)$ is a critical value, $U$ is a neighborhood of $u$ and $H_*$ denotes singular homology with $\mathbb{Z}_2$-coefficients.
My question concerns the "$H_*$ denotes singular homology with $\mathbb{Z}_2$-coefficients". How to explain in words what does it mean? I am not interested in technicalities, I just want to understand how it works and what it is. Please, use simplest concepts as possible.
I hope someone of you could help me to understand.
Thank you in advance!
${\bf EDIT:}$ Here $J^c=\{u\in E: J(u)\le c\}$, where $E$ denotes the inner product space.
 A: Singular homology groups are the most fundamental algebraic invariant of topological spaces. A traditional but terrible metaphor claims that the singular homology groups measure the presence of holes in a space, since for instance $H_n(S^n),$ where $S^n$ is the $n$-dimensional sphere, is $\mathbb Z,$ corresponding to an "$n$-dimensional hole" in the $n$-dimensional sphere. Here you are seeing singular homology with $\mathbb Z/2$ coefficients, which gives a vector space over the field with two elements measuring, perhaps, something about holes in your space without respect for orientation.
ADDED: The groups in your question are relative homology groups. If $A$ is a subspace of $B$ then $H_n(B,A)$ measures something about holes that would be in $B$ if $A$ were treated as a single point. In particular, the homology of an $n$-dimensional vector space relative to the complement of a point is the homology of an $n$-dimensional sphere discusses above--this is perhaps the simplest case of your situation, when $c$ is the global maximum of $J$ over $E.$
If this all seems aggressively vague, well, the definition and proof of basic properties of singular homology would usually take a month or so of a basic course in algebraic topology. You should learn this material. Singular homology (and other theories computing the same groups in the end) was a central motivation for the entire subject of algebraic topology, and it and its descendants and analogues are at the heart of every part of modern mathematics that has the slightest algebraic bent. Singular homology is thus something that every mathematician should learn.
