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\begin{document}

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\section*{Thom-Gysin sequence}

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

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

[[!include cohomology - contents]]

\hypertarget{algebraic_topology}{}\paragraph*{{Algebraic topology}}\label{algebraic_topology}

[[!include algebraic topology - contents]]

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Statement}{Statement}\dotfill \pageref*{Statement} \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}

The \emph{Thom-Gysin sequence} is a type of [[long exact sequence in cohomology]] induced by a [[spherical fibration]] and expressing the [[cohomology groups]] of the total space in terms of those of the base plus correction. The sequence may be obtained as a corollary of the [[Serre spectral sequence]] for the given fibration. It induces, and is induced by, the [[Thom isomorphism]].

\hypertarget{Statement}{}\subsection*{{Statement}}\label{Statement}

\begin{prop}
\label{ThomGysinSequence}\hypertarget{ThomGysinSequence}{}
Let $R$ be a [[commutative ring]] and let

\begin{displaymath}
\itexarray{
    S^n &\longrightarrow& E
    \\
    && \downarrow^{\mathrlap{\pi}}
    \\
    && B
  }
\end{displaymath}
be a [[Serre fibration]] over a [[simply connected topological space|simply connected]] [[CW-complex]] with typical [[fiber]] (\href{Introduction+to+Stable+homotopy+theory+--+P#FibersOfSerreFibrations}{exmpl.}) the [[n-sphere]].

Then there exists an element $c \in H^{n+1}(E; R)$ (in the [[ordinary cohomology]] of the total space with [[coefficients]] in $R$, called the \emph{[[Euler class]]} of $\pi$) such that the [[cup product]] operation $c \cup (-)$ sits in a [[long exact sequence]] of [[cohomology groups]] of the form

\begin{displaymath}
\cdots 
   \to 
   H^k(B; R) 
     \stackrel{\pi^\ast}{\longrightarrow}
   H^k(E; R)
    \stackrel{}{\longrightarrow}
   H^{k-n}(B;R)
   \stackrel{c \cup (-)}{\longrightarrow}
   H^{k+1}(B; R)
   \to
   \cdots
  \,.
\end{displaymath}
\end{prop}
(e.g. \hyperlink{Switzer75}{Switzer 75, section 15.30}, \hyperlink{Kochmann96}{Kochman 96, corollary 2.2.6})

\begin{proof}
Under the given assumptions there is the corresponding [[Serre spectral sequence]]

\begin{displaymath}
E_2^{s,t}
    \;=\;
  H^s(B; H^t(S^n;R))
    \;\Rightarrow\;
  H^{s+t}(E; R)
  \,.
\end{displaymath}
Since the [[ordinary cohomology]] of the [[n-sphere]] [[fiber]] is concentrated in just two degees

\begin{displaymath}
H^t(S^n; R)
  =
  \left\{
    \itexarray{
      R & for \; t= 0 \; and \; t = n
      \\
      0 & otherwise
    }
  \right.
\end{displaymath}
the only possibly non-vanishing terms on the $E_2$ page of this spectral sequence, and hence on all the further pages, are in bidegrees $(\bullet,0)$ and $(\bullet,n)$:

\begin{displaymath}
E^{\bullet,0}_2 \simeq H^\bullet(B; R)
  \,,
  \;\;\;\;
  and
  \;\;\;
  E^{\bullet,n}_2 \simeq H^\bullet(B; R)
  \,.
\end{displaymath}
As a consequence, since the differentials $d_r$ on the $r$th page of the Serre spectral sequence have bidegree $(r+1,-r)$, the only possibly non-vanishing differentials are those on the $(n+1)$-page of the form

\begin{displaymath}
\itexarray{
     E_{n+1}^{\bullet,n} & \simeq & H^\bullet(B;R)
     \\
     {}^{\mathllap{d_{n+1}}}\downarrow
     \\
     E_{n+1}^{\bullet+n+1,0} & \simeq &  H^{\bullet+n+1}(B;R)
  }
  \,.
\end{displaymath}
Now since the [[coefficients]] $R$ is a [[ring]], the [[Serre spectral sequence]] is [[multiplicative spectral sequence|multiplicative]] under [[cup product]] and the [[differential]] is a [[derivation]] (of total degree 1) with respect to this product. (See at \emph{\href{multiplicative+spectral+sequence#AHSSForMultiplicativeCohomology}{multiplicative spectral sequence -- Examples -- AHSS for multiplicative cohomology}}.)

To make use of this, write

\begin{displaymath}
\iota \coloneqq 1 \in H^0(B;R) \stackrel{\simeq}{\longrightarrow} E_{n+1}^{0,n}
\end{displaymath}
for the unit in the [[cohomology ring]] $H^\bullet(B;R)$, but regarded as an element in bidegree $(0,n)$ on the $(n+1)$-page of the spectral sequence. (In particular $\iota$ does \emph{not} denote the unit in bidegree $(0,0)$, and hence $d_{n+1}(\iota)$ need not vanish; while by the [[derivation]] property, it does vanish on the actual unit $1 \in H^0(B;R) \simeq E_{n+1}^{0,0}$.)

Write

\begin{displaymath}
c 
    \coloneqq 
  d_{n+1}(\iota) 
   \;\;
  \in E_{n+1}^{n+1,0} 
    \stackrel{\simeq}{\longrightarrow} 
  H^{n+1}(B; R)
\end{displaymath}
for the image of this element under the differential. We will show that this is the Euler class in question.

To that end, notice that every element in $E_{n+1}^{\bullet,n}$ is of the form $\iota \cdot b$ for $b\in E_{n+1}^{\bullet,0} \simeq H^\bullet(B;R)$.

(Because the [[multiplicative spectral sequence|multiplicative structure]] gives a group homomorphism $\iota \cdot(-) \colon H^\bullet(B;R) \simeq E_{n+1}^{0,0} \to E^{0,n}_{n+1} \simeq H^\bullet(B;R)$, which is an isomorphism because the product in the spectral sequence does come from the [[cup product]] in the [[cohomology ring]], see for instance \hyperlink{Kochmann96}{Kochman 96, first equation in the proof of prop. 4.2.9}, and since hence $\iota$ does act like the unit that it is in $H^\bullet(B;R)$).

Now since $d_{n+1}$ is a graded [[derivation]] and vanishes on $E_{n+1}^{\bullet,0}$ (by the above degree reasoning), it follows that its action on any element is uniquely fixed to be given by the product with $c$:

\begin{displaymath}
\begin{aligned}
    d_{n+1}(\iota \cdot b)
      & =
    d_{n+1}(\iota) \cdot b + (-1)^{n}\, \iota \cdot \underset{= 0}{\underbrace{d_{n+1}(b)}}
    \\
    & = c \cdot b
  \end{aligned}
  \,.
\end{displaymath}
This shows that $d_{n+1}$ is identified with the cup product operation in question:

\begin{displaymath}
\itexarray{
     E_{n+1}^{s,n} & \simeq & H^s(B;R)
     \\
     {}^{\mathllap{d_{n+1}}}\downarrow && \downarrow^{\mathrlap{c \cup (-)}}
     \\
     E_{n+1}^{s+n+1, 0} & \simeq & H^{s+n+1}(B;R)
  }
  \,.
\end{displaymath}
In summary, the non-vanishing entries of the $E_\infty$-page of the spectral sequence sit in [[exact sequences]] like so

\begin{displaymath}
\itexarray{
     0
     \\
     \downarrow
     \\
     E_\infty^{s,n}
     \\
     {}^{\mathllap{ker(d_{n+1})}}\downarrow
     \\
     E_{n+1}^{s,n} & \simeq & H^s(B;R)
     \\
     {}^{\mathllap{d_{n+1}}}\downarrow && \downarrow^{\mathrlap{c \cup (-)}}
     \\
     E_{n+1}^{s+n+1, 0} & \simeq & H^{s+n+1}(B;R)
     \\
     {}^{\mathllap{coker(d_{n+1})}}\downarrow
     \\
     E_\infty^{s+n+1,0}
     \\
     \downarrow
     \\
     0
  }  
  \,.
\end{displaymath}
Finally observe (lemma \ref{ImplicationsOfSparesnessOfSSSForSphericalFibration}) that due to the sparseness of the $E_\infty$-page, there are also [[short exact sequences]] of the form

\begin{displaymath}
0 \to E_\infty^{s,0} \longrightarrow H^s(E; R) \longrightarrow  E_\infty^{s-n,n} \to 0
  \,.
\end{displaymath}
Concatenating these with the above exact sequences yields the desired [[long exact sequence]].

\end{proof}
\begin{lemma}
\label{ImplicationsOfSparesnessOfSSSForSphericalFibration}\hypertarget{ImplicationsOfSparesnessOfSSSForSphericalFibration}{}
Consider a cohomology [[spectral sequence]] converging to some [[filtered object|filtered]] [[graded abelian group]] $F^\bullet C^\bullet$ such that

\begin{enumerate}%
\item $F^0 C^\bullet = C^\bullet$;


\item $F^{s} C^{\lt s} = 0$;


\item $E_\infty^{s,t} = 0$ unless $t = 0$ or $t = n$,



\end{enumerate}
for some $n \in \mathbb{N}$, $n \geq 1$. Then there are [[short exact sequences]] of the form

\begin{displaymath}
0 \to E_\infty^{s,0} \overset{}{\longrightarrow} C^s \longrightarrow  E_\infty^{s-n,n} \to 0
  \,.
\end{displaymath}
\end{lemma}
(e.g. \hyperlink{Switzer75}{Switzer 75, p. 356})

\begin{proof}
By definition of convergence of a spectral sequence, the $E_{\infty}^{s,t}$ sit in [[short exact sequences]] of the form

\begin{displaymath}
0 \to F^{s+1}C^{s+t} \overset{i}{\longrightarrow} F^s C^{s+t} \longrightarrow E_\infty^{s,t} \to 0
  \,.
\end{displaymath}
So when $E_\infty^{s,t} = 0$ then the morphism $i$ above is an [[isomorphism]].

We may use this to either shift away the filtering degree

\begin{itemize}%
\item if $t \geq n$ then $F^s C^{s+t} = F^{(s-1)+1}C^{(s-1)+(t+1)} \underoverset{\simeq}{i^{s-1}}{\longrightarrow} F^0 C^{(s-1)+(t+1)} = F^0 C^{s+t} \simeq C^{s+t}$;

\end{itemize}
or to shift away the offset of the filtering to the total degree:

\begin{itemize}%
\item if $0 \leq t-1 \leq n-1$ then $F^{s+1}C^{s+t} = F^{s+1}C^{(s+1)+(t-1)} 
\underoverset{\simeq}{i^{-(t-1)}}{\longrightarrow} F^{s+t}C^{(s+1)+(t-1)} = F^{s+t}C^{s+t}$

\end{itemize}
Moreover, by the assumption that if $t \lt 0$ then $F^{s}C^{s+t} = 0$, we also get

\begin{displaymath}
F^{s}C^{s} \simeq E_\infty^{s,0}
  \,.
\end{displaymath}
In summary this yields the vertical isomorphisms

\begin{displaymath}
\itexarray{
  0 
    &\to& 
  F^{s+1}C^{s+n} 
    &\longrightarrow& 
  F^{s}C^{s+n} 
    &\longrightarrow& 
  E_\infty^{s,n} 
    &\to& 
  0
  \\
  && 
    {}^{\mathllap{i^{-(n-1)}}}\downarrow^{\mathrlap{\simeq}} 
  &&
    {}^{\mathllap{i^{s-1}}}\downarrow^{\mathrlap{\simeq}}
  &&
    \downarrow^{\mathrlap{=}}
  \\
  0 
    &\to& 
  F^{s+n}C^{s+n}
  \simeq
  E_\infty^{s+n,0} 
    &\longrightarrow& 
  C^{s+n} 
    &\longrightarrow& 
  E_\infty^{s,n}
    &\to& 
  0
  }
\end{displaymath}
and hence with the top sequence here being exact, so is the bottom sequence.

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

\begin{itemize}%
\item [[Gysin map]]


\item [[Serre long exact sequence]]


\item [[Serre spectral sequence]]



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

\begin{itemize}%
\item [[Robert Switzer]], section 15.30 of \emph{Algebraic Topology - Homotopy and Homology}, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975 (\href{https://link.springer.com/book/10.1007/978-3-642-61923-6}{doi:10.1007/978-3-642-61923-6})


\item [[Stanley Kochmann]], section 2.2. of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996


\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Gysin_sequence}{Gysin sequence}}


\item [[James Milne]], section 16 of \emph{[[Lectures on Étale Cohomology]]}



\end{itemize}
Formalization in [[homotopy type theory]] is discussed in

\begin{itemize}%
\item [[Guillaume Brunerie]], \emph{On the homotopy groups of spheres in homotopy type theory} (\href{http://arxiv.org/abs/1606.05916}{arXiv:1606.05916})

\end{itemize}
Applications:

\begin{itemize}%
\item [[Martin Saralegi]], \emph{A Gysin Sequence for Semifree Actions of $S^3$}, Proceedings of the American Mathematical Society Vol. 118, No. 4 (Aug., 1993), pp. 1335-1345 (\href{http://www.jstor.org/stable/2160096}{jstor})

\end{itemize}
[[!redirects Thom-Gysin sequences]]

[[!redirects Thom-Gysin exact sequence]] [[!redirects Thom-Gysin exact sequences]]

[[!redirects Gysin sequence]] [[!redirects Gysin sequences]]

[[!redirects Gysin exact sequence]] [[!redirects Gysin exact sequences]]



\end{document}