Hopf algebra structure on cohomology

Definition

Let $k$ be a field and let $(X, e, \mu)$ be a path-connected H-space with the homotopy type of a CW complex. Assume that each $H^n(X; k)$ is finite-dimensional. By the Künneth theorem, the cross product gives an isomorphism $H^*(X; k) \otimes_k H^*(X; k) \cong H^*(X \times X; k)$. The Hopf algebra structure on cohomology of $X$ consists of:

1. the cohomology ring $H^*(X; k)$, with the cup product as multiplication and $1$ as unit;
2. the comultiplication $\Delta \colon H^*(X; k) \to H^*(X; k) \otimes_k H^*(X; k)$, defined as $\mu^*$ followed by the inverse of the Künneth isomorphism;
3. the counit $H^*(X; k) \to k$ given by restriction to the basepoint $e$.

With these maps, $H^0(X; k) = k$, $\Delta$ is a counital homomorphism of graded algebras, and $\Delta(x) = x \otimes 1 + 1 \otimes x + \sum x'_i \otimes x''_i$ with $0 < \abs{x'_i}, \abs{x''_i} < \abs{x}$ for $\abs{x} > 0$. If $\mu$ is homotopy associative, $\Delta$ is also coassociative and $H^*(X; k)$ is a connected, graded-commutative Hopf algebra.