\subsection{Compactly supported cohomology and the cohomology of

the mapping cone}\label{mappingconesection}

\vskip 0.5in

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.

\vskip 0.5in

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}

\vskip 0.5in

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 $H_2(X,\partial X,\Z)$.

\vskip 0.5in