Finitely generated cohomology Hopf algebras and torsion (Q1919893)

Let \(p\) be a prime. We begin a study of H-spaces whose cohomology mod \(p\) is finitely generated as an algebra. Such cohomology algebras are Hopf algebras and hence have Borel decompositions into tensor products of exterior algebras, truncated polynomial algebras and free polynomial algebras. The existence of \(p\)-torsion in the integral cohomology \(H^*(X; Z)\) plays a role in determining if the mod \(p\) cohomology \(H^*(X; Z_p)\) is finite-dimensional. In the following theorems, let \(X\) be a simply connected H-space whose mod \(p\) cohomology is finitely generated as an algebra. We prove: Theorem A. Generators of infinite height in the Borel decomposition lie in degrees \(2p^j\), for \(j \geq 0\). Theorem B. If \(X\) is two-connected and \(p^rH^*(X;Z)\) is torsion free for some \(r \geq 1\), then \(H^*(X; Z_p)\) is finite dimensional. Theorem C. Let \(X\) be two-connected. If \(H^* (X;Z_p)\) is not finite dimensional, then \(H^*(X;Z)\) has \(p\)-torsion of all orders. Theorem D. Let \(X\) be homotopy associative with \(H^*(X; Z_p)\) primitively generated. If \(p\) is an odd prime, then all even generators lie in degrees \(2p^j\), \(j \geq 0\) and for every even generator \(x\) of degree greater than 2, \(\beta_1 x \neq 0\).