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\begin{document}

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\section*{Boardman homomorphism}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{from_stable_cohomotopy_to_ordinary_cohomology}{From stable cohomotopy to ordinary cohomology}\dotfill \pageref*{from_stable_cohomotopy_to_ordinary_cohomology} \linebreak
\noindent\hyperlink{ForComplexOrientedCohomologyTheories}{For complex oriented cohomology theories}\dotfill \pageref*{ForComplexOrientedCohomologyTheories} \linebreak
\noindent\hyperlink{examples_2}{Examples}\dotfill \pageref*{examples_2} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Given any [[homotopy commutative ring spectrum]] $(E, \mu, e)$, then the \emph{Boardman homomorphism} is the [[homomorphism]] from [[stable homotopy groups]] (hence from [[stable homotopy homology theory]]) to $E$-[[generalized homology]] groups that is induced by [[smash product]] with the unit map $e \colon \mathbb{S} \longrightarrow E$ from the [[sphere spectrum]]:

\begin{displaymath}
\pi_\bullet(-)
  \simeq
  \pi_\bullet(\mathbb{S} \wedge (-))
  \longrightarrow
  \pi_\bullet(E \wedge (-))
  =
  E_\bullet(-)
  \,.
\end{displaymath}
For $E = H \mathbb{Z}$ the [[Eilenberg-MacLane spectrum]] for [[ordinary homology]], then this reduces to the [[Hurewicz homomorphism]] $\pi_\bullet(-) \to H_\bullet(-)$.

Dually, there is the Boardman homomorphism from [[stable cohomotopy]] to [[generalized cohomology]] induced under forming [[mapping spectra]] into the unit map of $E$:

\begin{displaymath}
\pi^\bullet(-)
  \simeq
  \pi_\bullet([(-),\mathbb{S}])
   \longrightarrow
  \pi_\bullet([(-),E])
   =
  E^\bullet(-)
  \,.
\end{displaymath}
Unifying these two cases, there is the bivariant Boardman homomorphism

\begin{displaymath}
[X, Y]_\bullet \simeq [X, Y \wedge \mathbb{S}]_\bullet \longrightarrow [X,Y \wedge E]_\bullet
  \,.
\end{displaymath}
Since [[generalized homology]]/[[generalized cohomology]] is typically more tractable than [[homotopy groups]]/[[cohomotopy]] (in particular when [[homology spectra split]]), the Boardman homomorphism is often used to partially reduce computations of the latter in terms of computations of the former.

One example is the computation of the homotopy grous of [[MU]] via the [[homology of MU]] ([[Quillen's theorem on MU]]), see \hyperlink{ForComplexOrientedCohomologyTheories}{below}.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{from_stable_cohomotopy_to_ordinary_cohomology}{}\subsubsection*{{From stable cohomotopy to ordinary cohomology}}\label{from_stable_cohomotopy_to_ordinary_cohomology}

Consider the [[unit]] morphism

\begin{displaymath}
\mathbb{S} \longrightarrow H \mathbb{Z}
\end{displaymath}
from the [[sphere spectrum]] to the [[Eilenberg-MacLane spectrum]] of the [[integers]]. For any [[topological space]]/[[spectrum]] postcomposition with this morphism induces [[Boardman homomorphisms]] of [[cohomology groups]] (in fact of [[commutative rings]])

\begin{equation}
b^n
  \;\colon\;
  \pi^n(X)
  \longrightarrow
  H^n(X, \mathbb{Z})
\label{BoardmandCohomotopyToOrdinaryCohomology}\end{equation}
from the [[stable cohomotopy]] of $X$ in degree $n$ to its [[ordinary cohomology]] in degree $n$.

\begin{prop}
\label{BoundsOnCoKernelOfBoardmandFromStableCohomotopyToOrdinaryCohomology}\hypertarget{BoundsOnCoKernelOfBoardmandFromStableCohomotopyToOrdinaryCohomology}{}
\textbf{(bounds on ([[cokernel|co-]])[[kernel]] of [[Boardman homomorphism]] from [[stable cohomotopy]] to [[integral cohomology]])}

If $X$ is a [[CW-spectrum]] which

\begin{enumerate}%
\item is [[n-connected object of an (infinity,1)-topos|(m-1)-connected]]


\item of dimension $d \in \mathbb{N}$



\end{enumerate}
then

\begin{enumerate}%
\item the [[kernel]] of the [[Boardman homomorphism]] $b^n$ \eqref{BoardmandCohomotopyToOrdinaryCohomology} for

\begin{displaymath}
m \leq n\leq d -1
\end{displaymath}
is a $\overline{\rho}_{d-n}$-[[torsion subgroup|torsion group]]:

\begin{displaymath}
\overline{\rho}_{d-n} ker(b^n) \;\simeq\; 0
\end{displaymath}

\item the [[cokernel]] of the [[Boardman homomorphism]] $b^n$ \eqref{BoardmandCohomotopyToOrdinaryCohomology} for

\begin{displaymath}
m \leq n  \leq d - 2
\end{displaymath}
is a $\overline{\rho}_{d-n-1}$-[[torsion subgroup|torsion group]]:

\begin{equation}
\overline{\rho}_{d-n-1} coker(b^n) \;\simeq\; 0
\label{TorsionEstimateCokernel}\end{equation}


\end{enumerate}
where

\begin{displaymath}
\overline{\rho}_{i}
   \;\coloneqq\;
   \left\{
   \itexarray{
     1 &\vert& i\leq 1
     \\
     \underoverset{j = 1}{i}{\prod}  
       exp\left( 
         \pi_j\left( \mathbb{S}\right)
       \right)
     &\vert&
     \text{otherwise}
   }
   \right.
\end{displaymath}
is the [[multiplication|product]] of the [[exponent of a group|exponents]] of the [[stable homotopy groups of spheres]] in [[positive number|positive]] degree $\leq i$.

\end{prop}
(\hyperlink{Arlettaz04}{Arlettaz 04, theorem 1.2})

\begin{example}
\label{ExampleForEstimatesOfTorsionOfCokernelOfBeta}\hypertarget{ExampleForEstimatesOfTorsionOfCokernelOfBeta}{}
\textbf{(estimates for torsion of cokernel of Boardman homomorphism)}

Let $X$ be a [[manifold]]

\begin{itemize}%
\item of [[dimension]] $d = 6$


\item [[simply connected topological space|simply connected]] with $\pi_2(X) \neq 0$



\end{itemize}
then Prop. \ref{BoundsOnCoKernelOfBoardmandFromStableCohomotopyToOrdinaryCohomology} asserts that the [[cokernel]] of the [[Boardman homomorphism]]

\begin{displaymath}
\beta^4
  \;\colon\;
  \mathbb{S}^4(X)
  \longrightarrow
  H^4( X, \mathbb{Z} )
\end{displaymath}
in

\begin{itemize}%
\item degree $n = 4$

\end{itemize}
is [[torsion subgroup|2-torsion]]:

\begin{displaymath}
2 coker(\beta^4) \;=\; 0
  \,.
\end{displaymath}
This is because in this case \eqref{TorsionEstimateCokernel} gives that the relevant torsion degree is

\begin{displaymath}
\begin{aligned}
    \overline{\rho}_{d-n-1}
    & =
    \overline{\rho}_{1}
    \\
    & = \exp( \pi_1(\mathbb{S}) )
    \\
    & = \exp( \mathbb{Z}/2 )     
    \\  
    & = 2
  \end{aligned}
  \,.
\end{displaymath}
Similarly, if instead the manifold has dimension $d = 7$ but sticking to degree $n = 4$, then the estimate is that the cokernel is [[torsion subgroup|4-torsion]],

\begin{displaymath}
4 coker(\beta^4) \;=\; 0
  \,.
\end{displaymath}
since then

\begin{displaymath}
\begin{aligned}
    \overline{\rho}_{d-n-1}
    & =
    \overline{\rho}_{2}
    \\
    & = \exp( \pi_1(\mathbb{S}) ) \cdot \exp( \pi_2(\mathbb{S}) )
    \\
    & = \exp( \mathbb{Z}/2 ) \cdot \exp( \mathbb{Z}/2 ) 
    \\  
    & = 2 \cdot 2
    \\
    & = 4
  \end{aligned}
  \,.
\end{displaymath}
Next for $d = 8$ we

\begin{displaymath}
\begin{aligned}
    \overline{\rho}_{d-n-1}
    & =
    \overline{\rho}_{3}
    \\
    & = 
      \exp( \pi_1(\mathbb{S}) ) 
      \cdot \exp( \pi_2(\mathbb{S}) )
      \cdot \exp( \pi_3(\mathbb{S}) )
    \\
    & = \exp( \mathbb{Z}/2 ) 
      \cdot \exp( \mathbb{Z}/2 ) 
      \cdot \exp( \mathbb{Z}/{24} ) 
    \\  
    & = 2 \cdot 2 \cdot 6
    \\
    & = 24
  \end{aligned}
  \,.
\end{displaymath}
\end{example}
\hypertarget{ForComplexOrientedCohomologyTheories}{}\subsubsection*{{For complex oriented cohomology theories}}\label{ForComplexOrientedCohomologyTheories}

Used for [[complex oriented cohomology theories]] and proof of [[Quillen's theorem on MU]] via the [[homology of MU]] (\ldots{})

(\hyperlink{Adams74}{Adams 74, pages 60-62}, \hyperlink{Lurie10}{Lurie 10, lecture 7})

\hypertarget{examples_2}{}\subsection*{{Examples}}\label{examples_2}

\begin{prop}
\label{BoardmanIsoOn7SphereMod2I}\hypertarget{BoardmanIsoOn7SphereMod2I}{}
\textbf{([[Boardman homomorphism|Boardman]] [[isomorphism]] on [[2-sphere]] mod [[binary icosahedral group]])}

Consider the [[binary icosahedral group]] $2 I$ and its [[action]] on the [[7-sphere]] induced via the identification $S^7 \simeq S(\mathbb{H} \times \mathbb{H})$ from the [[diagonal action|diagonal]] of the canonical action of $2I$ on the [[quaternions]] $\mathbb{H}$ induced via it being a [[finite subgroup of SU(2)]].

On the [[quotient space]] $S^7/2 I$ the [[Boardman homomorphism]] in degree 4 is an [[isomorphism]]

\begin{displaymath}
\mathbb{S}^4\left(
     S^7/2I
  \right)
  \underoverset{\simeq}{\beta}{\longrightarrow}
  H^4\left( S^7/2I , \mathbb{Z} \right)
\end{displaymath}
from [[stable cohomotopy]] in degree 4 to [[integral cohomology]] in degree 4.

\end{prop}
\begin{proof}
In terms of the [[Atiyah-Hirzebruch spectral sequence]] for [[stable cohomotopy]] it is sufficient to see that the two differentials

\begin{displaymath}
H^4\left(
    S^7/2I, \pi^0_s = \pi^s_0 =\mathbb{Z}
  \right)
  \overset{d_3}{\longrightarrow}
  H^6\left(
    S^7/2I, \pi^{-1}_s = \pi^s_{1} =\mathbb{Z}/2
  \right)
\end{displaymath}
and

\begin{displaymath}
H^4\left(
    S^7/2I, \pi^0_s = \pi^s_0 =\mathbb{Z}
  \right)
  \overset{d_3}{\longrightarrow}
  H^7\left(
    S^7/2I, \pi^{-2}_s = \pi^s_{2} =\mathbb{Z}/2
  \right)
\end{displaymath}
both vanish (all higher differentials on $H^4(-,\pi^0_s)$ vanish simply for dimensional reasons as $S^7$ is of [[dimension]] 7, while there are no differentials into $H^4(-,\pi^0_s)$ simply because the [[sphere spectrum]] is [[connective spectrum|connective]], so that the [[stable homotopy groups of spheres]] vanish in [[negative number|negative]] degree).

For $d_2$ to vanish, it is sufficient that

\begin{displaymath}
H^6\left(
    S^7/2I, \pi^{-1}_s = \pi^s_{1} =\mathbb{Z}/2
  \right)
  \;\simeq\;
  0
\end{displaymath}
We now first show that this is the case:

First, by the [[Gysin sequence]] for the [[spherical fibration]]

\begin{displaymath}
\itexarray{
    S^7 &\longrightarrow& S^7/SI
    \\
    && \downarrow
    \\
    && B (2 I)
  }
\end{displaymath}
we have

\begin{displaymath}
H^6\left(
    S^7/2I, \, \mathbb{Z}/2
  \right)
  \;\simeq\;
  H^6\left(
    B(2I),\, \mathbb{Z}/2
  \right)
  \,,
\end{displaymath}
where $B (2 I) \simeq \ast \sslash (2I)$ is the [[classifying space]] of $2I$ (see e.g. at [[infinity-action]]).

Moreover, by the [[universal coefficient theorem]] (\href{universal+coefficient+theorem#OrdinaryStatementInTopology}{this Prop.}) we have a [[short exact sequence]]

\begin{displaymath}
0 
    \to 
  Ext^1(H_{5}\big(B(2I), \mathbb{Z}), \mathbb{Z}/2\big) 
    \longrightarrow 
  H^6\big(B(2I), \mathbb{Z}/2\big)
    \longrightarrow 
  Hom_{Ab}\big( H_6( B(2I), \mathbb{Z}) , \mathbb{Z}/2 \big) 
    \to 
  0
  \,.
\end{displaymath}
This means that it is sufficient to see that

\begin{displaymath}
H_{5}\big(B(2I), \mathbb{Z}) \simeq 0
  \phantom{AAA}
  H_{6}\big(B(2I), \mathbb{Z}) \simeq 0
\end{displaymath}
But for every [[finite subgroup of SU(2)]] $G_{ADE} \subset SU(2)$ we have (by \href{https://ncatlab.org/nlab/show/finite+rotation+group#GroupCohomologyOfFiniteSubgroupsOfSU2}{this Prop.})

\begin{displaymath}
H_{5}\big(B(2I), \mathbb{Z}) \simeq G^{ab}_{ADE}
  \phantom{AAA}
  H_{6}\big(B(2I), \mathbb{Z}) \simeq 0
\end{displaymath}
where $G^{ab}_{ADE}$ is the [[abelianization]] of $G_{ADE}$. Specifically for $G_{ADE} = 2I$ this does vanish: the [[binary icosahedral group]] is a [[perfect group]] (\href{icosahedral+group#2IIsPerfect}{this Prop.}).

This shows that $d_2$ vanishes on $H^4(-, \pi^0)$.

Now by a standard argument, the AHSS-differentials between ordinary cohomology groups are stable [[cohomology operations]], and thus, if non-trivial, must be the [[Steenrod operations]] $Sq^n$ (e.g. \href{https://mathgroove.wordpress.com/2016/11/30/filling-the-details-5-two-words-about-the-atiyah-hirzebruch-spectral-sequence/}{here}, but let's add a more canonical reference).

This means first of all that if $d_2$ is not trivial then $d_2 = Sq^2$. But since that vanishes on $H^4(-,\pi^0)$ by the above argument, and on $H^7(-,\pi^2)$ for dimension reasons, so that the relevant entries pass as ordinary cohomology groups to the third page of the [[spectral sequence]], it follows similarly that $d_3 = Sq^3$.

But by the [[Adem relation]] $Sq^3 = Sq^1 \circ Sq^2$, the vanishing of $Sq_2$ on $H^4(-,\pi^0)$ then also implies the vanishing of $d_3$ on this entry.

\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Hopf degree theorem]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Named after [[Michael Boardman]].

\begin{itemize}%
\item [[John Frank Adams]], part II, section 6 of \emph{[[Stable homotopy and generalised homology]]} (1974)


\item [[Stanley Kochmann]], section 4.3 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996


\item Dominique Arlettaz, \emph{The generalized Boardman homomorphisms}, Central European Journal of Mathematics March 2004, Volume 2, Issue 1, pp 50-56


\item [[Jacob Lurie]], lecture 7 of \emph{[[Chromatic Homotopy Theory]]}, 2010, (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture7.pdf}{pdf})


\item [[Akhil Mathew]], \emph{Torsion exponents in stable homotopy and the Hurewicz homomorphism}, Algebr. Geom. Topol. 16 (2016) 1025-1041 (\href{https://arxiv.org/abs/1501.07561}{arXiv:1501.07561})



\end{itemize}
[[!redirects Boardman homomorphisms]]



\end{document}