tone down splitting odd primary splitting results

Author: ecpeterson

unknowns.tex

@@ -40,7 +40,7 @@ \subsection*{Splitting bordism spectra}
     \item First, one calculates the homology \(\HFtwo_* MX\) of the bordism spectrum as a comodule over the dual Steenrod algebra.
     \item Then, one shows that the associated sheaf \((\HFtwo_* MX)\widetilde{\quad}\) is the pushforward of a simpler sheaf for some smaller structure group \(G \le \InternalAut_1(\G_a)\).  (This amounts to understanding the Steenrod subcomodule generated by the unit class.)
     \item Finally, one demonstrates some algebraic structure theorem for sheaves arising through such pushforwards.  These are referred to generally as ``Milnor--Moore-type theorems''.
-    \item These results then assemble to yield a splitting at the level of \(2\)--complete spectra through Adams spectral sequence methods.\footnote{Replacing \(2\) by \(p\) everywhere, one can also produce odd-primary results.}
+    \item These results then assemble to yield a splitting at the level of \(2\)--complete spectra through Adams spectral sequence methods.\footnote{Replacing \(2\) by \(p\) everywhere, one can also produce less dramatic odd-primary results.}
 \end{enumerate}
 
 For instance, by studying the homology of \(M\SO\), one finds that the unit class in \(\HFtwo_* M\SO\) generates the submodule~\cite[Lemma 20.38]{Switzer} \[1 \cdot \mathcal A_* \mmod (\Sq^1) \subseteq \HFtwo_* M\SO,\] and an appropriate generalization of Milnor--Moore then gives a splitting \[M\SO_{(2)} \simeq \left( \bigvee_j \Susp^{n_j} \HFtwo \right) \vee \left( \bigvee_k \Susp^{m_k} H\Z \right).\]  Similarly, the homology of \(M\Spin\) reveals an inclusion~\cite{ABS,ABP,GiambalvoPengelley} \[1 \cdot \mathcal A_* \mmod (\Sq^1, \Sq^2) \subseteq \HFtwo_* M\Spin,\] and a further generalization of Milnor--Moore ultimately produces a splitting\footnote{However, this time the splitting is not one of ring spectra.  In fact, further wild nonmultiplicative splittings are available~\cite{BordismSplitting}.} \[M\Spin_{(2)} \simeq \left( \bigvee_j kO[n_j, \infty)\right) \vee \left( \bigvee_k \Susp^{m_k} H\F_2 \right).\]  When studying \(M\String\), one finds that there is an inclusion \[1 \cdot \mathcal A_* \mmod (\Sq^1, \Sq^2, \Sq^4) \subseteq \HFtwo_* M\String,\] but at this point the Adams spectral sequence becomes too intricate to analyze effectively and no analogous splitting of \(M\String\) has yet been produced.\footnote{See, however, recent work of Laures--Schuster~\cite[Section 2]{LauresSchuster}.}\footnote{There are also \(p\)--local complex analogues of these results: