loek.tex

\subsection{Compactly supported cohomology and the cohomology of
the mapping cone}\label{mappingconesection}

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We briefly review the mapping-cone-complex realization of the
cohomology of compact supports of $X$. For a more detailed discussion,
see \cite{FMspec}, section~5.

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We let $A_c^{\bullet}(X)$ be the complex of compactly supported
differential forms on $X$ which gives rise to $H_c^{\bullet}(X)$,
the cohomology of compact supports. We now represent the
compactly-supported cohomology of $X$ by the cohomology of the
mapping cone $C^{\bullet}$ of $i^*$, see \cite{Weibel}, p.19, where
as before $i: X \hookrightarrow \overline{X}$. However, we will
change the sign of the differential on $C^{\bullet}$ and shift the
grading down by one. Thus we have $$C^i =\{ (a,b), a \in A^i
(\overline{X}), b \in A^{i-1}(\partial \overline{X})\}$$ with $d(a,b)
= (da, i^*a - db)$.  If $(a,b)$ is a cocycle in $C^{\bullet}$ we
will use $[[a,b]]$ to denote its cohomology class. We have
\begin{proposition} \label{quasiiso} The cochain map $A_c^{\bullet}(X)
\to C^{\bullet}$ given by $c \mapsto (c,0)$ is a quasi-isomorphism.
\end{proposition}

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We now give a cochain map from $C^{\bullet}$ to $ A_c^{\bullet}(X)$
which induces the inverse to the above isomorphism. We let $V$ be
a product neighborhood of $\partial \overline{X}$ as in
Section~\ref{BScomp}, and we let $\pi:V \to \partial \overline{X}$
be the projection. If $b$ is a form on $\partial \overline{X}$ we
obtain a form $\pi^{\ast} b$ on V. Let $f$ be a smooth function of
the geodesic flow coordinate $t$ which is $1$ near $t=\infty$ and
zero for $t \leq T$ for some sufficiently large $T$. We may regard
$f$ as a function on $V$ by making it constant on the $\partial
\overline{X}$ factor. We extend $f$ to all of $ \overline{X}$ by
making it zero off of $V$. Let $(a,b)$ be a cocycle in $C^i$. Then
there exist a compactly supported closed form $\alpha $ and a form
$\mu$ which vanishes on $\partial \overline{X}$ such that \[ a -
d(f \pi^{\ast}b ) = \alpha + d\mu.  \] We define the cohomology
class $[a,b]$ in the compactly supported  cohomology $H^i_c(X)$ to
be the class of $\alpha$, and the assignment $[[a,b]] \mapsto [a,b]$
gives the desired inverse. From this we obtain the  following
integral formulas for the Kronecker pairings with $[a,b]$.  \vskip
0.5in \begin{lemma}\label{integralformula} Let $\eta$ be a closed
form on $\overline{X}$ and $C$ a relative cycle in $\overline{X}$
of appropriate degree. Then $$ \langle[a, b], [\eta]]\rangle =
\int_{\overline{X}}a\wedge \eta - \int_{\partial \overline{X}} b
\wedge i^*\eta, \ \text{and} \ \ \langle [a,b],C \rangle =  \int_{C}a
- \int_{\partial C} b.$$ \end{lemma} \vskip 0.5in \section{Capped
special cycles and linking numbers in Sol}\label{capped-cycles}

For $x \in V$ such that $(x,x)>0$, we define \[ D_x =\{ z \in D;
\, z \perp x \}.  \] Then $D_x$ is an embedded upper half plane in
$D$. We let $\G_x \subset \G$ be the stabilizer of $x$ and define
the special or Hirzebruch-Zagier cycle by \[ C_x = \G_x \back D_x,
\] and by slight abuse identify $C_x$ with its image in $X$. These
are modular or Shimura curves. For positive $n \in \Q$, we write
$\calL_n = \{ x \in \mathcal{L}; \, \tfrac12(x,x)= n\}$. Then the
composite cycles $C_n$ are given by \[ C_n= \sum_{x \in \G \back
\calL_n} C_x.  \] Since the divisors define in general relative
cycles, we take the sum in $H_2(X,\partial X,\Z)$.


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