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Let me begin by a couple of questions :

Consider a graded abelian group $V=\oplus_{i\geq 0} V_i$ such that $V_{2i}=\mathbb{Z}$ and $V_{\textrm{odd}}=0$. What are the possible Hopf algebra structures on it?

One can ask a slightly stronger question :

When does a given Hopf algebra structure on $V$ (as above) arise as the integral cohomology of a $H$-space?

The motivation behind this question is purely my own curiosity. While discussing how to distinguish $\Omega S^3$ and $\mathbb{CP}^\infty$ rationally, we saw that the rational cohomology or the rational homotopy groups are unable to detect the difference. However, $H^\ast(\Omega S^3;\mathbb{Z})=\Gamma_{\mathbb{Z}}[\alpha]$, the divided polynomial algebra generated by $\alpha$ (of degree $2$) while $H^\ast(\mathbb{CP}^\infty;\mathbb{Z})=\mathbb{Z}[u]$ is the polynomial algebra generated by $u$ (of degree $2$). Moreover, a polynomial algebra such as $\mathbb{Z}[u]$ has a comultiplication map given by $u\stackrel{\Delta}{\longrightarrow}1\otimes u+u\otimes 1$ and extended naturally. One can check that the dual (as a Hopf algebra) of $\mathbb{Z}[u]$ is isomorphic to $\Gamma_{\mathbb{Z}}[u^\ast]$, where $u^\ast$ is the dual of $u$.

From what I could conclude by playing around with coefficients is that for each prime $p$ and a positive integer $r$ one can cook up a Hopf algebra structure on $V$. I don't know if they come from a space. However, these structure constants must be compatible with the action of the Steenrod algebra (or the mod $p$ version) if $V=H^\ast(X;\mathbb{Z})$ for some $H$-space $X$. I vaguely remember that compatibility with the Steenrod algebra is not sufficient and Adams operations provide further obstructions (although I may be wrong on this point). This leads me to :

Is there a (list of) necessary and sufficient criteria (in general or at least in this case) which tells us when a given Hopf algebra structure on graded vector space arises as $H^\ast(X;\mathbb{Z})$ for some $H$-space?

This may be well known (and perhaps classical) to homotopy theorists and any reference to known results are good enough for me.

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You will certainly be interested to read Chapter 3.C, "H-Spaces and Hopf Algebras" of Hatcher's "Algebraic Topology", although I don't know if the discussion there answers your question. (This book is available free online: math.cornell.edu/~hatcher/AT/ATpage.html) –  Tom Church Sep 21 '10 at 7:39

1 Answer 1

Call a generating class in degree 1 'x'. It is forced to be primitive. The identity $$ \Delta(x^n) = (1 \otimes x + x \otimes 1)^n \neq 0 $$ shows that, after tensoring with $\mathbb{Q}$, the resulting ring is a polynomial ring on your primitive class. Thus, you find that your Hopf algebra is some sub-Hopf algebra of $\mathbb{Q}[x]$ and contains $\mathbb{Z}[x]$.

In particular, in each degree $n$ there is a unique positive integer $a_n$ such that $x^n/a_n$ is a generator of $V_n$. We have $a_1 = 1$, and this subset being closed under multiplication is equivalent to $a_{n+m}/a_n a_m \in \mathbb{Z}$ for all $n$ and $m$.

Take duals. The rationalization of the dual is also a polynomial algebra on $x^*$, and $V_n^*$ is generated by $${a_n} (x^n)^* = \frac{a_n}{n!} (x^*)^n.$$ This subset of $\mathbb{Q}[x^*]$ being closed under multiplication is equivalent to $\binom{n+m}{n} \cdot \frac{a_n a_m}{a_{n+m}} \in \mathbb{Z}$ for all n and m.

The possible Hopf algebra structures are therefore defined by all such sequences of positive integers $a_n$ so that $a_1 = 1$ and $a_n a_m$ divides $a_{n+m}$ divides $\binom{n+m}{n} a_n a_m$.

The realizability problem is much harder and I do not have a concrete answer for you.

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This is what I referred to briefly in my question when I said "playing around with coefficients". In fact, for a prime $p$ and a positive integer $r$ if one defines $a_i=p^{[\frac{i}{p^r}]}$ then this passes the divisibility tests. However, I don't know if it arises from a space. –  Somnath Basu Sep 21 '10 at 13:55
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One thing you can do is use Sullivan's localization technology to take $\Omega S^3$ localized at one set of primes, together with $\mathbb{CP}^\infty$ localized at the complementary set of primes, and glue them together into one H-space whose (co)homology is formed by the same patching procedure. (And more generally you can realize the Hopf algebra one prime at a time.) I'm not sure about your example, however. –  Tyler Lawson Sep 21 '10 at 17:50

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