  In this paper we explore new relations between Algebraic Topology and the
theory of Hopf Algebras. For an arbitrary topological space $X$, the loop space
homology $H_*(\Omega\Sigma X; \coefZ)$ is a Hopf algebra. We introduce a new
homotopy invariant of a topological space $X$ taking for its value the
isomorphism class (over the integers) of the Hopf algebra $H_*(\Omega\Sigma X;
\coefZ)$. This invariant is trivial if and only if the Hopf algebra
$H_*(\Omega\Sigma X; \coefZ)$ is isomorphic to a Lie-Hopf algebra, that is, to
a primitively generated Hopf algebra. We show that for a given $X$ these
invariants are obstructions to the existence of a homotopy equivalence $\Sigma
X\simeq \Sigma^2Y$ for some space $Y$.
  Further on, using the notion of Hopf algebras, we establish new structural
properties of the cohomology ring, in particular, of the cup product. For
example, using the fact that the suspension of a polyhedral product $X$ is a
double suspension, we obtain a strong condition on the cohomology ring
structure of $X$. This gives an important application in toric topology. For an
algebra to be realised as the cohomology ring of a moment-angle manifold $\Z_P$
associated to a simple polytope $P$, we found an obstruction in the Hopf
algebra $H_*(\Omega\Sigma \Z_P)$.
  In addition, we use homotopy decompositions to study particular Hopf
algebras.
