closes #125

Author: ecpeterson

preface.tex

@@ -57,7 +57,7 @@ \subsection*{Acknowledgements}
 
 \subsection*{Changes since publication}
 
-It has now been some years since the publication of this text, and it has been very gratifying to see the uptick in readers' interest in this subject.  Of course, increased interest also brings with it corrections and addenda.  I would particularly like to thank Jonathan Beardsley, Robert Burklund, Shachar Carmeli, Jeremy Hahn, Yigal Kamel, Kiran Luecke, Jeroen van der Meer, Piotr Pstr\k{a}gowski, Charles Rezk, John Rognes, Bruno Stonek, Paul VanKoughnett, and Allen Yuan for their insightful questions and contributions to the continued health of this document.  While most of the changes have been minor, the major differences from the published version are:
+It has now been some years since the publication of this text, and it has been very gratifying to see the uptick in readers' interest in this subject.  Of course, increased interest also brings with it corrections and addenda.  I would particularly like to thank Jonathan Beardsley, Robert Burklund, Shachar Carmeli, Jeremy Hahn, Yigal Kamel, Kiran Luecke, Jeroen van der Meer, Piotr Pstr\k{a}gowski, Charles Rezk, John Rognes, Bruno Stonek, Timon Tausendpfund, Paul VanKoughnett, and Allen Yuan for their insightful questions and contributions to the continued health of this document.  While most of the changes have been minor, the major differences from the published version are:
 
 \begin{itemize}
     \item There were a pair of mutually cancelling errors in \Cref{StabilizingTheMUSteenrodOps}: the calculation leading up to \Cref{AjAndBjAreInTheFGLSubring} mistook the permutation representation for the reduced permutation representation, and so was missing a factor of \(x\); and \Cref{PCnOnUniversalMUClasses} was argued incorrectly, which hid this previous error.  These have both now been corrected.

quillen.tex

@@ -884,15 +884,24 @@ \section{Explicitly stabilizing cyclic \texorpdfstring{\(MU\)}{MU}--power operat
 \end{align*}
 and the fact that \(P^{C_n}\) is natural under pullback, it will suffice for us to study the effect of \(P^{C_n}\) on the universal classes \[u_m\co MU(m) \to MU,\] after reinterpreting them as classes \(\bar u_m\co T_m BU(m) \to \Susp^{2m} MU\) on a space.
 
-We begin with the canonical orientation itself, \(\bar u_1 \in MU^2 T_1 BU(1)\).  Our access to this class comes from relating it to \(MU\)--Chern classes: the complex analogue of \Cref{RPnThomExample} shows the zero-section map \(\zeta_1\co BU(1) \to T_1 BU(1)\) to be a reduced cohomology isomorphism which sends \(\bar u_1\) to the canonical coordinate \(x = c_1(\L) \in MU^2 \CP^\infty\), so that we can equivalently compute \(P^{C_n}(x)\).  Then, \(MU\)--Chern classes intertwine with the \(H_\infty^2\)--structure on \(MU\) according to the following diagram:
+We begin with the canonical orientation itself, \(\bar u_1 \in MU^2 T_1 BU(1)\).
+Our access to this class comes from relating it to \(MU\)--Chern classes: the complex analogue of \Cref{RPnThomExample} shows the zero-section map \(\zeta_1\co BU(1) \to T_1 BU(1)\) to be a reduced cohomology isomorphism which sends \(\bar u_1\) to the canonical coordinate \(x = c_1(\L) \in MU^2 \CP^\infty\), so that we can equivalently compute \(P^{C_n}(x)\).
+Then, \(MU\)--Chern classes intertwine with the \(H_\infty^2\)--structure on \(MU\) according to the following diagram:
 \begin{center}
 \begin{tikzcd}
 \Susp^\infty_+ BU(1) \arrow["\Delta"]{d} \arrow{r} & \Susp^\infty_+ BU(1) \sm \Susp^\infty_+ BC_n \arrow["\Delta_{hC_n}"]{d} \\
 (\Susp^\infty_+ BU(1))^{\sm n} \arrow{r} \arrow[bend right=15, "\L^{\times n}"' near start]{rr} \arrow["c_1^{\sm n}"]{d} & \Susp^\infty_+ BU(1)^{\sm n}_{hC_n} \arrow["\L(C_n)"]{r} \arrow[crossing over]{d} \arrow["P^{C_n}_{\mathrm{ext}}" near end, crossing over]{rd} & \Susp^\infty_+ BU(n) \arrow["c_n"]{d} \\
 (\Susp^2 MU)^{\sm n} \arrow{r} & (\Susp^2 MU)^{\sm n}_{hC_n} \arrow["\mu_{n,2}"]{r} & \Susp^{2n} MU.
 \end{tikzcd}
 \end{center}
-The commutativity of the widest rectangle (i.e., the justification for the name ``\(c_n\)'' on the right-most vertical arrow) comes from the Cartan formula for Chern classes: because \(\L^{\oplus n}\) splits as the sum of \(n\) line bundles, \(c_n(\L^{\oplus n})\) is computed as the product of the \(1\)\textsuperscript{st} Chern classes of those line bundles.  Second, the commutativity of the right-most square is not trivial: it is a specific consequence of how the multiplicative structure on \(MU\) arises from the direct sum of vector bundles.\footnote{In general, any notion of first Chern class \(\Susp^\infty_+ BU(1) \to \Susp^2 E\) gives rise to a \emph{noncommuting} diagram of this same shape.  The two composites \(\Susp^\infty_+ BU(1)^{\sm n}_{hC_n} \to \Susp^{2n} E\) need not agree, since \(\L(C_n)\) has no \textit{a priori} reason to be compatible with the factorization appearing in the \(H_\infty^2\)--structure.  They turn out to be loosely related nonetheless, and their exact relation (as well as a procedure for making them agree in certain cases) is the subject of \Cref{PowerOpnsSection}.}  The commutativities of the other two squares comes from the natural transformation from a \(C_n\)--space to its homotopy orbit space.
+The commutativity of the widest rectangle (i.e., the justification for the name ``\(c_n\)'' on the right-most vertical arrow) comes from the Cartan formula for Chern classes: because \(\L^{\oplus n}\) splits as the sum of \(n\) line bundles, \(c_n(\L^{\oplus n})\) is computed as the product of the \(1\)\textsuperscript{st} Chern classes of those line bundles.
+The commutativity of the right-most square follows from a calculation on Thom classes: monoidality of the Thom functor shows that the Thom class of $\L^{\times n}$ is the $n$--fold power of $\bar u_1$; a calculation on fibers shows that the Thom class $T(\L(C_n)) \to \Susp^{2n} MU$ descends to homotopy orbits; and the statement on top Chern classes follows by pulling back along the zero section.%
+\footnote{%
+In general, any notion of first Chern class \(\Susp^\infty_+ BU(1) \to \Susp^2 E\) gives rise to a \emph{noncommuting} diagram of this same shape.
+The two composites \(\Susp^\infty_+ BU(1)^{\sm n}_{hC_n} \to \Susp^{2n} E\) need not agree, since \(\L(C_n)\) has no \textit{a priori} reason to be compatible with the factorization appearing in the \(H_\infty^2\)--structure.
+They turn out to be loosely related nonetheless, and their exact relation (as well as a procedure for making them agree in certain cases) is the subject of \Cref{PowerOpnsSection}.
+}
+The commutativities of the other two squares comes from the natural transformation from a \(C_n\)--space to its homotopy orbit space.
 
 Hence, the internal cyclic power operation \(P^{C_n}(x)\) applied to the canonical coordinate \(x\) is defined by the composite \[\Susp^\infty_+ BU(1)_{hC_n} \xrightarrow{\Delta_{hC_n}} (\Susp^\infty_+ BU(1))^{\sm n}_{hC_n} \to \Susp^\infty_+ BU(n) \to \Susp^{2n} MU,\] which is to say \[P^{C_n}(x) = c_n(\Delta_{hC_n}^* \L(C_n)).\]  We have thus reduced to computing a particular Conner--Floyd--Chern class of a particular bundle.  Our next move is to transport more information from the \(C_n\)--equivariant bundle \(\Delta^* \L^{\times n}\) to the bundle \(\Delta_{hC_n}^* \L(C_n)\).