which by (13.5) and (13.6) satisfies
$\big{|}\beta[\mathscr{N}]\big{|}<\big{|}\varphi[\mathscr{N}]\big{|}$. So by
the minimality demand (0 ) there is a good pair $(\mathscr{X},\xi)$ resolving
$(\mathscr{N},\beta)$ and satisfying
$|\mathscr{X}|\leq\frac{|\mathscr{N}|-2}{g-1}$. In particular,
$(\mathscr{X},\xi)$ resolves $(\mathscr{N},\varphi)$, contrary to (0 ). This
whole argument shows $|\mathscr{A}^{\prime}|\geq 2$ and in a similar fashion
one confirms $|\mathscr{B}^{\prime}|\geq 2$ as well. Thereby (13.7) is proved.

For all these reasons (0 ) tells us that the good pairs
$(\mathscr{A}^{\prime},\alpha)$ and $(\mathscr{B}^{\prime},\beta)$ are
resolved by certain good pairs $(\mathscr{Y},\upsilon)$ and
$(\mathscr{Z},\zeta)$ that satisfy

$|\mathscr{Y}|\leq\frac{|\mathscr{A}^{\prime}|-2}{g-1}\quad\text{ and
}\quad|\mathscr{Z}|\leq\frac{|\mathscr{B}^{\prime}|-2}{g-1}\,,$ (13.8)

respectively. Set $\mathscr{X}=\mathscr{Y}\cup\mathscr{Z}$ and let
$\xi\subseteq\upsilon\cup\zeta$ be a map from $\mathscr{X}$ to
$\mathscr{H}^{+}$. We contend that

$\text{the good pair $(\mathscr{X},\xi)$ resolves $(\mathscr{N},\varphi)$}\,.$
(13.9)

The demand ( ) ‣ 13.2 is clear and ( ) ‣ 13.2 follows from

$\xi[\mathscr{X}]\subseteq\upsilon[\mathscr{Y}]\cup\zeta[\mathscr{Z}]\overset{\text{\ref{it:6149b}}}{\subseteq}\alpha[\mathscr{A}^{\prime}]\cup\beta[\mathscr{B}^{\prime}]\overset{\eqref{eq:6111}}{=}\varphi[\mathscr{A}]\cup\varphi[\mathscr{B}]=\varphi[\mathscr{N}]\,.$

For the verification of ( ) ‣ 13.2 we want to apply Lemma 13.6 to
$\mathscr{A}\cup\mathscr{Y}$, $\mathscr{B}\cup\mathscr{Z}$, and $x$ here in
place of $\mathscr{A}$, $\mathscr{B}$, and $e$ there. As the assumption that
$(\mathscr{A}\cup\mathscr{Y})\cup\\{x^{+}\\}=\mathscr{A}^{\prime}\cup\mathscr{Y}$
and
$(\mathscr{B}\cup\mathscr{Z})\cup\\{x^{+}\\}=\mathscr{B}^{\prime}\cup\mathscr{Z}$
need to be forests of copies are satisfied by our choice of the good pairs
$(\mathscr{Y},\upsilon)$ and $(\mathscr{Z},\zeta)$, it remains to check that

$V(\mathscr{A}\cup\mathscr{Y})\cap V(\mathscr{B}\cup\mathscr{Z})\subseteq
x\,.$ (13.10)

Towards the proof of this inclusion we observe that $V(F_{\star})\cap
V(H)\subseteq V\bigl{(}\varphi(F_{\star})\bigr{)}$ holds for every
$F_{\star}\in\mathscr{A}$, whence $V(\mathscr{A})\cap V(H)\subseteq
V(\varphi[\mathscr{A}])$. Similarly one has

$V(\mathscr{Y})\cap V(H)\subseteq
V(\upsilon[\mathscr{Y}])\overset{\text{\ref{it:6149b}}}{\subseteq}V(\alpha[\mathscr{A}^{\prime}])\overset{\eqref{eq:6111}}{=}V(\varphi[\mathscr{A}])$

and both statements together yield $V(\mathscr{A}\cup\mathscr{Y})\cap
V(H)\subseteq V(\varphi[\mathscr{A}])$. Proceeding similarly with
$\mathscr{B}\cup\mathscr{Z}$ and combining the results we infer

$\bigl{(}V(\mathscr{A}\cup\mathscr{Y})\cap
V(\mathscr{B}\cup\mathscr{Z})\bigr{)}\cap V(H)\subseteq
V(\varphi[\mathscr{A}])\cap V(\varphi[\mathscr{B}])=x\,.$ (13.11)

Moreover, the definition of the partite amalgamation (13.3) entails
$V(F_{\star})\cap V(F_{\star\star})\subseteq V(H)$ for all
$F_{\star}\in\mathscr{A}\cup\mathscr{Y}$ and
$F_{\star\star}\in\mathscr{B}\cup\mathscr{Z}$ and for this reason we have

$V(\mathscr{A}\cup\mathscr{Y})\cap V(\mathscr{B}\cup\mathscr{Z})\subseteq
V(H)\,.$

Together with (13.11) this establishes (13.10) and thus concludes the proof of
(13.9).

However, in the light of

$|\mathscr{X}|\leq|\mathscr{Y}|+|\mathscr{Z}|\overset{\eqref{eq:6141}}{\leq}\frac{|\mathscr{A}|-1}{g-1}+\frac{|\mathscr{B}|-1}{g-1}=\frac{|\mathscr{N}|-2}{g-1}$

this contradicts (0 ). In other words, the case that $x$ is an edge is
impossible.

Second Case. $|x|\leq 1$.

We argue similar as in the first case. The pairs
$(\mathscr{A},\varphi\upharpoonright\mathscr{A})$ and
$(\mathscr{B},\varphi\upharpoonright\mathscr{B})$ are good and the main point
is that they are resolved by certain good pairs $(\mathscr{Y},\upsilon)$ and
$(\mathscr{Z},\zeta)$ such that

$|\mathscr{Y}|\leq\frac{|\mathscr{A}|-1}{g-1}\quad\text{ and
}\quad|\mathscr{Z}|\leq\frac{|\mathscr{B}|-1}{g-1}\,,$ (13.12)

respectively. In fact, if $|\mathscr{A}|\geq 2$ we can argue exactly as in the
first case in order to obtain such a good pair $(\mathscr{Y},\upsilon)$
satisfying the stronger estimate
$|\mathscr{Y}|\leq\frac{|\mathscr{A}|-2}{g-1}$. Moreover, in case
$|\mathscr{A}|=1$ we can just take $\mathscr{Y}=\upsilon=\varnothing$. The
claim on $(\mathscr{Z},\zeta)$ is proved in the same way.

Again we set $\mathscr{X}=\mathscr{Y}\cup\mathscr{Z}$ and take a map
$\xi\subseteq\upsilon\cup\zeta$ from $\mathscr{X}$ to $\mathscr{H}^{+}$. We
remark that (13.10) is still valid in the present case and the only change we
need to make when concluding that $\mathscr{N}\cup\mathscr{X}$ is a forest of
copies is that this time we need to appeal to Lemma 13.5. Apart from these
small modifications, the proof that the good pair $(\mathscr{X},\xi)$ resolves
$(\mathscr{N},\varphi)$ is still the same. Finally, (13.12) implies
$|\mathscr{X}|\leq\frac{|\mathscr{N}|-2}{g-1}$, meaning that again we reach a
contradiction to the choice of $(\mathscr{N},\varphi)$ in (0 ). ∎

After these preparations we can quickly establish the main result of this
subsection.

###### Proof of Theorem 13.12.

Let $\Phi$ be any linear Ramsey construction and let $\Xi$ be a partite lemma
applicable to $k$-partite $k$-uniform hypergraphs $F$ with
$\mathrm{girth}(F)>n$ that delivers linear systems of hypergraphs
$(H,\mathscr{H})$ satisfying $\mathrm{Girth}(H,\mathscr{H}^{+})>n$. We recall
that the existence of such a partite lemma is an immediate consequence of
Theorem 10.19.

Let us attempt to perform the partite construction
$\mathrm{PC}(\Phi,\Xi)_{r}(F)=(H,\mathscr{H})$. By Lemma 13.14 picture zero is
$(g,n)$-certified and an iterative application of Lemma 13.15 discloses that
all further pictures are well-defined and, likewise, $(g,n)$-certified. In
particular, the final picture is well-defined and the fact that it is
$(g,n)$-certified implies that it has the desired property. ∎

Finally, we deduce the last statement announced in the introduction.

###### Proof of Theorem 1.6.

Given a linear $k$-uniform hypergraph $F$ and $r,n\in{\mathds{N}}$ we apply
Theorem 1.7 with $n^{\prime}=\binom{n}{k}$ instead of $n$, thus obtaining some
linear system $(H,\mathscr{H})$. Without loss of generality we can assume that
$E(H)=E(\mathscr{H})$, for deleting edges from $H$ that belong to none of the
copies in $\mathscr{H}$ cannot influence whether the system $(H,\mathscr{H})$
satisfies the conclusion of Theorem 1.7.

In order to show that $H$ has the desired property we consider any set
$X\subseteq E(H)$ whose size it at most $n$. For every edge $e$ of $H$
contained in $X$ we fix some copy $F_{e}\in\mathscr{H}$ such that $e\in
E(F_{e})$. Since there are at most $\binom{|X|}{k}$ such edges, the set

$\mathscr{N}^{-}=\bigl{\\{}F_{e}\in\mathscr{H}\colon e\in E(H)\text{ and
}e\subseteq X\bigr{\\}}$

has at most the size $n^{\prime}$. In the special case that
$|\mathscr{N}^{-}|\leq 1$ we can set $\mathscr{N}=\mathscr{N}^{-}$ and are
done. Otherwise the conclusion of Theorem 1.7 yields a set
$\mathscr{X}\subseteq\mathscr{H}$ such that
$\mathscr{N}=\mathscr{N}^{-}\cup\mathscr{X}$ is a forest of copies with the
desired property. ∎

### Acknowledgement

We would like to thank Joanna Polcyn for drawing the beautiful pictures and
Mathias Schacht for devoting a great amount of his time to reading and
studying this article with us while we have been writing it.

Moreover, we thank Mathias Schacht for writing an initial draft of Section 2.
The current version owes a lot to his efforts.

## Index

  * $(F,\equiv^{F})\ltimes(X,W)$ Example 11.4, Definition 6.6
  * $(g,n)$-certified Definition 13.13
  * $\diamondsuit$ Definition 10.12
  * $A$-intersecting hypergraph §10.3, Definition 3.15
  * acceptable big cycle Definition 8.3
  * admissible enumeration §13.1
  * amenable partite lemma Definition 12.5
  * big cycle (in a system of pretrains) Definition 8.1
  * clean intersection Definition 3.3
  * clean partite lemma §3.4, Corollary 3.12, Lemma 3.17, §5.1, §7.4
  * collapse Definition 4.8
  * conductor §9.1
  * connector Definition 4.3, item ($B2$)
  * constant coordinate §3.4, §7.1
  * constituent §3.2
  * construction §3.3
  * contraction (of wagons) §6.1
  * cycle of copies Definition 4.3
  * disjoint union (of quasitrains) §11.1
  * edge copy §4.2, §8.1
  * edge forest §13.1
  * extended linear system §4.2
  * extended system of pretrains §8.1
  * extension (of pretrains) Definition 6.2
  * extension process §6.3
  * $f$-partite hypergraph Definition 3.7
  * $f$-partite pretrain §6.1
  * $f$-partite subpretrains §6.1
  * forest of copies Definition 1.5, §13.1
  * $\mathrm{Girth}$ Definition 10.10, §4.2, Definition 8.9
  * $\mathfrak{girth}$ Definition 10.7, Definition 8.16
  * $h(\mathscr{C})$ §4.2
  * Hales-Jewett construction §3.1, Lemma 3.16, §7.1
  * Hales-Jewett theorem §3.1
  * height Definition 10.1
  * horizontal §3.2
  * induced Ramsey theorem §3.3
  * induced subpretrain §6.1
  * induction along partite construction §3.4
  * initial copy §13.1
  * length §4.2
  * linear construction §3.7
  * linear partite lemma §3.3, §3.7
  * linear pretrain Definition 6.9
  * linear quasitrain Definition 10.7
  * linear system (of hypergraphs) §3.7
  * linear system of pretrains Definition 6.9
  * living Definition 9.2
  * long piece Definition 8.4
  * master copy Definition 4.8
  * mixed index item $\bullet$, item $\bullet$
  * moving coordinate §3.4, §7.1
  * music line §3.2
  * $n$-cycle Definition 1.3
  * Nešetřil §1.2, §3, §6
  * order §4.2, §8.1
  * ordered hypergraph §3.5
  * ordered hypergraph pair Definition 6.5
  * ordered pretrain §6.1
  * ordered subpretrain §6.1
  * parameter Definition 10.4
  * partial $F$-forest Definition 1.5
  * partite amalgamation §3.2
  * partite lemma §3.1, §3.3
  * picture §3.2
  * picture zero §3.2
  * piece Definition 8.4
  * pretrain §6.1
  * pretrain picture §7.2
  * pretrain picture zero §7.2
  * product Ramsey theorem §3.6
  * pure index item $\bullet$, item $\bullet$
  * quasitrain Definition 10.1
  * quasitrain picture §12.1
  * quasitrain system §10.1
  * Ramsey construction for $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern 0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern 0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$-trains Definition 12.5
  * Ramsey’s theorem §3
  * real copy §4.2, §8.1
  * reduct §9.2
  * revisability Definition 12.9
  * scattered Definition 9.11
  * semitidy Definition 4.17
  * set system §4.1
  * short piece Definition 8.4
  * standard copy (in a pretrain extension) §6.2
  * standard copy (in partite constructions) §3.2
  * strongly induced Definition 3.1
  * subpretrain Definition 6.1
  * subquasitrain Definition 10.6
  * subtrain Definition 10.6
  * supreme copy Definition 8.5
  * system of hypergraphs §3.3
  * system of pretrains §6.1
  * tame extension Definition 6.7
  * terminal copy Definition 13.1
  * tidy Definition 4.6
  * train Definition 10.4
  * train picture Definition 12.1
  * transitivity of girth §4.1
  * twin §9.1
  * vanishing (of wagons) §6.1
  * vertical §3.2
  * wagon §6.1
  * wagon assimilation Example 11.3, §6.2
  * $\mu$-wagon Definition 10.1
  * $\Omega^{(2)}$ §10.3, §11.2, §3.4, Corollary 3.14, Lemma 3.18, Proposition 3.8, Corollary 5.6, §7.4

## References


# Early thermalization of quark-gluon matter initially created in high-energy
nucleus-nucleus collisions

Xiao-Ming Xu  Department of Physics, Shanghai University, Baoshan, Shanghai
200444, China<EMAIL_ADDRESS>
###### Abstract

Elastic parton-parton-parton scattering is briefly reviewed and is included in
transport equations of quark-gluon matter. We solve the transport equations
and get thermal states from initially produced quark-gluon matter. Both gluon
matter and quark matter take early thermalization, but gloun matter has a
shorter thermalization time than quark matter.

##  1 Introduction

A large amount of particles move randomly and momentum distribution functions
in different directions are identical. The distribution depends only on the
absolute value of the three-dimensional momentum. The dependence indicates the
existence of a thermal state and exposes the basic quantity temperature, which
value is acquired from the analytic expression of the distribution (e.g.,
temperature is in the Boltzmann distribution). Unfortunately, the system of
gluons, quarks and antiquarks produced in initial nucleus-nucleus collisions
at the Relativistic Heavy Ion Collider (RHIC) energies does not possess
temperature. This is because the distribution in the incoming nucleus beam
direction is much larger than the distribution in the direction perpendicular
to the beam direction [1]. However, the system evolves rapidly into a thermal
state (a quark-gluon plasma) as concluded from the elliptic flow data of
hadrons [2] and the corresponding explanation of hydrodynamic calculations
which assume early thermalization and ideal relativistic fluid flow [3, 4].
The early thermalization or the rapid creation of temperature is generally
interesting. Parton-parton scattering was employed to study thermalization of
initially produced quark-gluon matter, but is not enough to explain the early
thermalization [5, 6, 7, 8]. Elastic gluon-gluon-gluon scattering was proposed
to explain the early thermalization of gluon matter [9]. Nevertheless, elastic
quark-quark scattering and elastic quark-quark-quark scattering give a long
thermalization time of quark matter [10]. Recently we have obtained that quark
matter takes the early thermalization due to all types of elastic scattering
that involves quarks [11]. In the next section I briefly introduce the elastic
scattering of quark-quark-quark [10], quark-quark-antiquark [12], quark-
antiquark-antiquark, antiquark-antiquark-antiquark, gluon-quark-quark [13],
gluon-quark-antiquark [13], gluon-antiquark-antiquark, gluon-gluon-quark [11],
gluon-gluon-antiquark or gluon-gluon-gluon [9]. In section 3 we present
transport equations that include all types of elastic parton-parton-parton
scattering and numerical results of the equations. Conclusions are in the last
section.

Table 1: The number of diagrams for elastic parton-parton-parton scattering, the number of diagrams in a class, the number of the triple-gluon vertex and the number of the four-gluon vertex of a diagram. elastic | number of | class | number of | number of | number of
---|---|---|---|---|---
scattering | diagrams |  | diagrams | the 3-gluon | the 4-gluon
| for a type |  | in a class | vertex | vertex
quark-quark-quark 1 | 42 | 1 | 36 | 0 | 0
|  | 2 | 6 | 1 | 0
quark-quark-quark 2 | 14 | 1 | 12 | 0 | 0
|  | 2 | 2 | 1 | 0
quark-quark-quark 3 | 7 | 1 | 6 | 0 | 0
|  | 2 | 1 | 1 | 0
quark-quark-antiquark 1 | 58 | 1 | 52 | 0 | 0
|  | 2 | 6 | 1 | 0
quark-quark-antiquark 2 | 14 | 1 | 12 | 0 | 0
|  | 2 | 2 | 1 | 0
quark-quark-antiquark 3 | 29 | 1 | 26 | 0 | 0
|  | 2 | 3 | 1 | 0
quark-quark-antiquark 4 | 7 | 1 | 6 | 0 | 0
|  | 2 | 1 | 1 | 0
gluon-quark-quark 1 | 72 | 1 | 40 | 0 | 0
|  | 2 | 24 | 1 | 0
|  | 3 | 8 | 2 | 0
or |  |  |  | 0 | 1
gluon-quark-quark 2 | 36 | 1 | 20 | 0 | 0
|  | 2 | 12 | 1 | 0
|  | 3 | 4 | 2 | 0
or |  |  |  | 0 | 1
gluon-quark-antiquark 1 | 76 | 1 | 40 | 0 | 0
|  | 2 | 28 | 1 | 0
|  | 3 | 8 | 2 | 0
or |  |  |  | 0 | 1
gluon-quark-antiquark 2 | 36 | 1 | 20 | 0 | 0
|  | 2 | 12 | 1 | 0
|  | 3 | 4 | 2 | 0
or |  |  |  | 0 | 1
gluon-gluon-quark | 123 | 1 | 24 | 0 | 0
|  | 2 | 36 | 1 | 0
|  | 3 | 38 | 2 | 0
or |  |  |  | 0 | 1
|  | 4 | 25 | 3 | 0
or |  |  |  | 1 | 1
gluon-gluon-gluon | 220 | 1 | 105 | 4 | 0
|  | 2 | 96 | 2 | 1
|  | 3 | 19 | 0 | 2

##  2 Elastic parton-parton-parton scattering

Each type of elastic 3-to-3 scattering is divided into 2, 3 or 4 classes
according to the number of the triple-gluon vertex and the number of the four-
gluon vertex. In Table 1 we list the number of diagrams for each type or in a
class, the number of the triple-gluon vertex and the number of the four-gluon
vertex of a diagram. The number following the entry quark-quark-quark has the
meanings: the three quarks are identical at 1; only two quarks are identical
at 2; the three quarks are different at 3. The number following the entry
quark-quark-antiquark has the meanings: the two quarks and the antiquark have
the same flavor at 1; only the two quarks have the same flavor at 2; only one
quark has the same flavor as the antiquark at 3; the three flavors of the two
quarks and the antiquark are not identical at 4. The number following the
entry gluon-quark-quark (gluon-quark-antiquark) has the meanings: the two
quarks (the quark and the antiquark) have the same flavor at 1; the two quarks
(the quark and the antiquark) take different flavors at 2.

Feynman diagrams for the elastic quark-antiquark-antiquark scattering
(antiquark-antiquark-antiquark, gluon-antiquark-antiquark, gluon-gluon-
antiquark) are obtained from the Feynman diagrams for the elastic quark-quark-
antiquark (quark-quark-quark, gluon-quark-quark, gluon-gluon-quark) via the
replacement of quark by antiquark and vice versa. To illustrate elastic 3-to-3
scattering, we show four Feynman diagrams in figures 1-4. In figures 1 and 2
quark-antiquark annihilation and creation occur. In both figure 3 and figure 4
two triple-gluon vertices and one four-gluon vertex are involved.

Figure 1: Elastic gluon-quark-antiquark scattering. The wiggly (solid) lines
stand for gluons (quarks and antiquarks).

Figure 2: Same as figure 1.

Figure 3: Elastic gluon-gluon-gluon scattering. The wiggly lines stand for
gluons.

Figure 4: Same as figure 3.

The elastic gluon-gluon-gluon scattering is complicated but rich in the
process as shown by the 220 Feynman diagrams at the tree level [9]. To better
understand the scattering, we sort out Feynman diagrams in every class. We
select three triple-gluon vertices of a diagram to subclassify the diagrams in
the first class and six subclasses are obtained according to the initial
gluons and the final gluons possessed by the selected vertices. The subclasses
are listed in Table 2. Every diagram in the second class has a four-gluon
vertex and two triple-gluon vertices. We select the four-gluon vertex and a
triple-gluon vertex to subclassify the diagrams in the second class and nine
subclasses are obtained according to the initial gluons and the final gluons
possessed by the four-gluon vertex and the selected triple-gluon vertex. The
subclasses are listed in Table 3. Diagrams in the third class can not be
subclassified.

Table 2: Number of diagrams in a subclass identified by the three selected triple-gluon vertices. $n_{1\rm i}$ and $n_{1\rm f}$ ($n_{2\rm i}$ and $n_{2\rm f}$, $n_{3\rm i}$ and $n_{3\rm f}$) are numbers of initial and final gluons possessed by the first (second, third) triple-gluon vertex, respectively. subclass | number of | $n_{1\rm i}$ | $n_{1\rm f}$ | $n_{2\rm i}$ | $n_{2\rm f}$ | $n_{3\rm i}$ | $n_{3\rm f}$
---|---|---|---|---|---|---|---
| diagrams |  |  |  |  |  |
1 | 36 | 1 | 1 | 1 | 1 | 1 | 0
2 | 18 | 0 | 2 | 1 | 1 | 1 | 0
3 | 18 | 2 | 0 | 1 | 1 | 0 | 1
4 | 18 | 2 | 0 | 0 | 2 | 1 | 0
5 | 6 | 1 | 1 | 1 | 1 | 0 | 0
6 | 9 | 2 | 0 | 0 | 2 | 0 | 0
Table 3: Number of diagrams in a subclass identified by the four-gluon vertex and the selected triple-gluon vertex. $n_{{\rm i}4}$ and $n_{{\rm f}4}$ ($n_{{\rm i}3}$ and $n_{{\rm f}3}$) are numbers of initial and final gluons possessed by the 4-gluon vertex (the 3-gluon vertex), respectively. subclass | number of | $n_{{\rm i}4}$ | $n_{{\rm f}4}$ | $n_{{\rm i}3}$ | $n_{{\rm f}3}$
---|---|---|---|---|---
| diagrams |  |  |  |
1 | 18 | 1 | 2 | 1 | 1
2 | 18 | 2 | 1 | 1 | 1
3 | 9 | 1 | 2 | 2 | 0
4 | 9 | 2 | 1 | 0 | 2
5 | 3 | 0 | 3 | 2 | 0
6 | 3 | 3 | 0 | 0 | 2
7 | 18 | 1 | 1 | 1 | 1
8 | 9 | 0 | 2 | 2 | 0
9 | 9 | 2 | 0 | 0 | 2

##  3 Transport equations and thermal states

We assume that quark-gluon matter consists of gluons, quarks and antiquarks
with up and down flavors and the two flavors (the quark and the antiquark as
well) have the same amount. Let the distribution functions for the gluon, the
up-quark, the down-quark, the up-antiquark and the down-antiquark be $f_{{\rm
g}i}$, $f_{ui}$, $f_{di}$, $f_{\bar{u}i}$ and $f_{\bar{d}i}$, respectively,
where $i$ labels the $i$th parton in scattering, and
$f_{ui}=f_{di}=f_{\bar{u}i}=f_{\bar{d}i}=f_{qi}$. The transport equation for
gluon matter is

$\displaystyle\frac{\partial f_{{\rm g}1}}{\partial t}$ $\displaystyle+$
$\displaystyle\vec{\rm v}_{1}\cdot\vec{\nabla}_{\vec{r}}f_{{\rm
g}1}~{}~{}=~{}~{}-\frac{1}{2E_{1}}\int\frac{d^{3}p_{2}}{(2\pi)^{3}2E_{2}}\frac{d^{3}p_{3}}{(2\pi)^{3}2E_{3}}\frac{d^{3}p_{4}}{(2\pi)^{3}2E_{4}}(2\pi)^{4}\delta^{4}(p_{1}+p_{2}-p_{3}-p_{4})$
$\displaystyle\times\left\\{\frac{{\rm g}_{G}}{2}\mid{\cal M}_{{\rm g}{\rm
g}\to{\rm g}{\rm g}}\mid^{2}[f_{{\rm g}1}f_{{\rm g}2}(1+f_{{\rm
g}3})(1+f_{{\rm g}4})-f_{{\rm g}3}f_{{\rm g}4}(1+f_{{\rm g}1})(1+f_{{\rm
g}2})]\right.$ $\displaystyle+{\rm g}_{Q}(\mid{\cal M}_{{\rm g}u\to{\rm
g}u}\mid^{2}+\mid{\cal M}_{{\rm g}d\to{\rm g}d}\mid^{2}+\mid{\cal M}_{{\rm
g}\bar{u}\to{\rm g}\bar{u}}\mid^{2}+\mid{\cal M}_{{\rm g}\bar{d}\to{\rm
g}\bar{d}}\mid^{2})$ $\displaystyle\left.\times[f_{{\rm g}1}f_{q2}(1+f_{{\rm
g}3})(1-f_{q4})-f_{{\rm g}3}f_{q4}(1+f_{{\rm g}1})(1-f_{q2})]\right\\}$
$\displaystyle-\frac{1}{2E_{1}}\int\frac{d^{3}p_{2}}{(2\pi)^{3}2E_{2}}\frac{d^{3}p_{3}}{(2\pi)^{3}2E_{3}}\frac{d^{3}p_{4}}{(2\pi)^{3}2E_{4}}\frac{d^{3}p_{5}}{(2\pi)^{3}2E_{5}}\frac{d^{3}p_{6}}{(2\pi)^{3}2E_{6}}$
$\displaystyle\times(2\pi)^{4}\delta^{4}(p_{1}+p_{2}+p_{3}-p_{4}-p_{5}-p_{6})\left\\{\frac{{\rm
g}_{G}^{2}}{12}\mid{\cal M}_{{\rm g}{\rm g}{\rm g}\to{\rm g}{\rm g}{\rm
g}}\mid^{2}\right.$ $\displaystyle\times[f_{{\rm g}1}f_{{\rm g}2}f_{{\rm
g}3}(1+f_{{\rm g}4})(1+f_{{\rm g}5})(1+f_{{\rm g}6})-f_{{\rm g}4}f_{{\rm
g}5}f_{{\rm g}6}(1+f_{{\rm g}1})(1+f_{{\rm g}2})(1+f_{{\rm g}3})]$
$\displaystyle+\frac{{\rm g}_{G}{\rm g}_{Q}}{2}(\mid{\cal M}_{{\rm g}{\rm
g}u\to{\rm g}{\rm g}u}\mid^{2}+\mid{\cal M}_{{\rm g}{\rm g}d\to{\rm g}{\rm
g}d}\mid^{2}+\mid{\cal M}_{{\rm g}{\rm g}\bar{u}\to{\rm g}{\rm
g}\bar{u}}\mid^{2}+\mid{\cal M}_{{\rm g}{\rm g}\bar{d}\to{\rm g}{\rm
g}\bar{d}}\mid^{2})$ $\displaystyle\times[f_{{\rm g}1}f_{{\rm
g}2}f_{q3}(1+f_{{\rm g}4})(1+f_{{\rm g}5})(1-f_{q6})-f_{{\rm g}4}f_{{\rm
g}5}f_{q6}(1+f_{{\rm g}1})(1+f_{{\rm g}2})(1-f_{q3})]$ $\displaystyle+{\rm
g}_{Q}^{2}[\frac{1}{4}\mid{\cal M}_{{\rm g}uu\to{\rm
g}uu}\mid^{2}+\frac{1}{2}(\mid{\cal M}_{{\rm g}ud\to{\rm
g}ud}\mid^{2}+\mid{\cal M}_{{\rm g}du\to{\rm
g}du}\mid^{2})+\frac{1}{4}\mid{\cal M}_{{\rm g}dd\to{\rm g}dd}\mid^{2}$
$\displaystyle+\mid{\cal M}_{{\rm g}u\bar{u}\to{\rm
g}u\bar{u}}\mid^{2}+\mid{\cal M}_{{\rm g}u\bar{d}\to{\rm
g}u\bar{d}}\mid^{2}+\mid{\cal M}_{{\rm g}d\bar{u}\to{\rm
g}d\bar{u}}\mid^{2}+\mid{\cal M}_{{\rm g}d\bar{d}\to{\rm g}d\bar{d}}\mid^{2}$
$\displaystyle+\frac{1}{4}\mid{\cal M}_{{\rm g}\bar{u}\bar{u}\to{\rm
g}\bar{u}\bar{u}}\mid^{2}+\frac{1}{2}(\mid{\cal M}_{{\rm
g}\bar{u}\bar{d}\to{\rm g}\bar{u}\bar{d}}\mid^{2}+\mid{\cal M}_{{\rm
g}\bar{d}\bar{u}\to{\rm g}\bar{d}\bar{u}}\mid^{2})+\frac{1}{4}\mid{\cal
M}_{{\rm g}\bar{d}\bar{d}\to{\rm g}\bar{d}\bar{d}}\mid^{2}]$
$\displaystyle\left.\times[f_{{\rm g}1}f_{q2}f_{q3}(1+f_{{\rm
g}4})(1-f_{q5})(1-f_{q6})-f_{{\rm g}4}f_{q5}f_{q6}(1+f_{{\rm
g}1})(1-f_{q2})(1-f_{q3})]\right\\},$

and the transport equation for up-quark matter is

$\displaystyle\frac{\partial f_{q1}}{\partial t}$ $\displaystyle+$
$\displaystyle\vec{\rm
v}_{1}\cdot\vec{\nabla}_{\vec{r}}f_{q1}~{}~{}=~{}~{}-\frac{1}{2E_{1}}\int\frac{d^{3}p_{2}}{(2\pi)^{3}2E_{2}}\frac{d^{3}p_{3}}{(2\pi)^{3}2E_{3}}\frac{d^{3}p_{4}}{(2\pi)^{3}2E_{4}}(2\pi)^{4}\delta^{4}(p_{1}+p_{2}-p_{3}-p_{4})$
$\displaystyle\times\left\\{{\rm g}_{G}\mid{\cal M}_{u{\rm g}\to u{\rm
g}}\mid^{2}[f_{q1}f_{{\rm g}2}(1-f_{q3})(1+f_{{\rm g}4})-f_{q3}f_{{\rm
g}4}(1-f_{q1})(1+f_{{\rm g}2})]\right.$ $\displaystyle+{\rm
g}_{Q}(\frac{1}{2}\mid{\cal M}_{uu\to uu}\mid^{2}+\mid{\cal M}_{ud\to
ud}\mid^{2}+\mid{\cal M}_{u\bar{u}\to u\bar{u}}\mid^{2}+\mid{\cal
M}_{u\bar{d}\to u\bar{d}}\mid^{2})$
$\displaystyle\times\left.[f_{q1}f_{q2}(1-f_{q3})(1-f_{q4})-f_{q3}f_{q4}(1-f_{q1})(1-f_{q2})]\right\\}$
$\displaystyle-\frac{1}{2E_{1}}\int\frac{d^{3}p_{2}}{(2\pi)^{3}2E_{2}}\frac{d^{3}p_{3}}{(2\pi)^{3}2E_{3}}\frac{d^{3}p_{4}}{(2\pi)^{3}2E_{4}}\frac{d^{3}p_{5}}{(2\pi)^{3}2E_{5}}\frac{d^{3}p_{6}}{(2\pi)^{3}2E_{6}}$
$\displaystyle\times(2\pi)^{4}\delta^{4}(p_{1}+p_{2}+p_{3}-p_{4}-p_{5}-p_{6})\left\\{\frac{{\rm
g}_{G}^{2}}{4}\mid{\cal M}_{u{\rm g}{\rm g}\to u{\rm g}{\rm
g}}\mid^{2}\right.$ $\displaystyle\times[f_{q1}f_{{\rm g}2}f_{{\rm
g}3}(1-f_{q4})(1+f_{{\rm g}5})(1+f_{{\rm g}6})-f_{q4}f_{{\rm g}5}f_{{\rm
g}6}(1-f_{q1})(1+f_{{\rm g}2})(1+f_{{\rm g}3})]$ $\displaystyle+{\rm
g}_{Q}{\rm g}_{G}(\frac{1}{2}\mid{\cal M}_{uu{\rm g}\to uu{\rm
g}}\mid^{2}+\mid{\cal M}_{ud{\rm g}\to ud{\rm g}}\mid^{2}+\mid{\cal
M}_{u\bar{u}{\rm g}\to u\bar{u}{\rm g}}\mid^{2}+\mid{\cal M}_{u\bar{d}{\rm
g}\to u\bar{d}{\rm g}}\mid^{2})$ $\displaystyle\times[f_{q1}f_{q2}f_{{\rm
g}3}(1-f_{q4})(1-f_{q5})(1+f_{{\rm g}6})-f_{q4}f_{q5}f_{{\rm
g}6}(1-f_{q1})(1-f_{q2})(1+f_{{\rm g}3})]$ $\displaystyle+{\rm
g}_{Q}^{2}[\frac{1}{12}\mid{\cal M}_{uuu\to uuu}\mid^{2}+\frac{1}{4}(\mid{\cal
M}_{uud\to uud}\mid^{2}+\mid{\cal M}_{udu\to
udu}\mid^{2})+\frac{1}{4}\mid{\cal M}_{udd\to udd}\mid^{2}$
$\displaystyle+\frac{1}{2}\mid{\cal M}_{uu\bar{u}\to
uu\bar{u}}\mid^{2}+\frac{1}{2}\mid{\cal M}_{uu\bar{d}\to
uu\bar{d}}\mid^{2}+\mid{\cal M}_{ud\bar{u}\to ud\bar{u}}\mid^{2}+\mid{\cal
M}_{ud\bar{d}\to ud\bar{d}}\mid^{2}$ $\displaystyle+\frac{1}{4}\mid{\cal
M}_{u\bar{u}\bar{u}\to u\bar{u}\bar{u}}\mid^{2}+\frac{1}{2}(\mid{\cal
M}_{u\bar{u}\bar{d}\to u\bar{u}\bar{d}}\mid^{2}+\mid{\cal
M}_{u\bar{d}\bar{u}\to u\bar{d}\bar{u}}\mid^{2})+\frac{1}{4}\mid{\cal
M}_{u\bar{d}\bar{d}\to u\bar{d}\bar{d}}\mid^{2}]$
$\displaystyle\times\left.[f_{q1}f_{q2}f_{q3}(1-f_{q4})(1-f_{q5})(1-f_{q6})-f_{q4}f_{q5}f_{q6}(1-f_{q1})(1-f_{q2})(1-f_{q3})]\right\\},$

where $\rm\vec{v}_{1}$ is the parton velocity; ${\rm g}_{G}$ (${\rm g}_{Q}$)
is the gluon (quark) color-spin degeneracy factor;
$p_{i}(i=1,\cdot\cdot\cdot,6)$ denote the four-momenta of initial and final
partons; $E_{i}$ is the energy component of $p_{i}$. In equation (1) ${\cal
M}_{gd\bar{d}\to gd\bar{d}}$ is the amplitude for the elastic scattering of a
gluon, a down-quark and a down-antiquark; in equation (2) ${\cal
M}_{u\bar{d}\bar{u}\to u\bar{d}\bar{u}}$ is the amplitude for the elastic
scattering of an up-quark, a down-antiquark and an up-antiquark. Other
notations can be similarly understood. The transport equations for down-quark
matter, up-antiquark matter and down-antiquark matter can be established just
as the equation for up-quark matter.

We take the following replacement in the transport equations

$f_{{\rm g}1}f_{{\rm g}2}(1+f_{{\rm g}3})(1+f_{{\rm g}4})-f_{{\rm g}3}f_{{\rm
g}4}(1+f_{{\rm g}1})(1+f_{{\rm g}2})\to f_{{\rm g}1}f_{{\rm g}2}-f_{{\rm
g}3}f_{{\rm g}4},$ $f_{{\rm g}1}f_{q2}(1+f_{{\rm g}3})(1-f_{q4})-f_{{\rm
g}3}f_{q4}(1+f_{{\rm g}1})(1-f_{q2})\to f_{{\rm g}1}f_{q2}-f_{{\rm
g}3}f_{q4},$ $f_{{\rm g}1}f_{{\rm g}2}f_{{\rm g}3}(1+f_{{\rm g}4})(1+f_{{\rm
g}5})(1+f_{{\rm g}6})-f_{{\rm g}4}f_{{\rm g}5}f_{{\rm g}6}(1+f_{{\rm
g}1})(1+f_{{\rm g}2})(1+f_{{\rm g}3})\to f_{{\rm g}1}f_{{\rm g}2}f_{{\rm
g}3}-f_{{\rm g}4}f_{{\rm g}5}f_{{\rm g}6},$ $f_{{\rm g}1}f_{{\rm
g}2}f_{q3}(1+f_{{\rm g}4})(1+f_{{\rm g}5})(1-f_{q6})-f_{{\rm g}4}f_{{\rm
g}5}f_{q6}(1+f_{{\rm g}1})(1+f_{{\rm g}2})(1-f_{q3})\to f_{{\rm g}1}f_{{\rm
g}2}f_{q3}-f_{{\rm g}4}f_{{\rm g}5}f_{q6},$ $f_{{\rm
g}1}f_{q2}f_{q3}(1+f_{{\rm g}4})(1-f_{q5})(1-f_{q6})-f_{{\rm
g}4}f_{q5}f_{q6}(1+f_{{\rm g}1})(1-f_{q2})(1-f_{q3})\to f_{{\rm
g}1}f_{q2}f_{q3}-f_{{\rm g}4}f_{q5}f_{q6},$ $f_{q1}f_{{\rm
g}2}(1-f_{q3})(1+f_{{\rm g}4})-f_{q3}f_{{\rm g}4}(1-f_{q1})(1+f_{{\rm g}2})\to
f_{q1}f_{{\rm g}2}-f_{q3}f_{{\rm g}4},$
$f_{q1}f_{q2}(1-f_{q3})(1-f_{q4})-f_{q3}f_{q4}(1-f_{q1})(1-f_{q2})\to
f_{q1}f_{q2}-f_{q3}f_{q4},$ $f_{q1}f_{{\rm g}2}f_{{\rm
g}3}(1-f_{q4})(1+f_{{\rm g}5})(1+f_{{\rm g}6})-f_{q4}f_{{\rm g}5}f_{{\rm
g}6}(1-f_{q1})(1+f_{{\rm g}2})(1+f_{{\rm g}3})\to f_{q1}f_{{\rm g}2}f_{{\rm
g}3}-f_{q4}f_{{\rm g}5}f_{{\rm g}6},$ $f_{q1}f_{q2}f_{{\rm
g}3}(1-f_{q4})(1-f_{q5})(1+f_{{\rm g}6})-f_{q4}f_{q5}f_{{\rm
g}6}(1-f_{q1})(1-f_{q2})(1+f_{{\rm g}3})\to f_{q1}f_{q2}f_{{\rm
g}3}-f_{q4}f_{q5}f_{{\rm g}6},$
$f_{q1}f_{q2}f_{q3}(1-f_{q4})(1-f_{q5})(1-f_{q6})-f_{q4}f_{q5}f_{q6}(1-f_{q1})(1-f_{q2})(1-f_{q3})\to
f_{q1}f_{q2}f_{q3}-f_{q4}f_{q5}f_{q6}.$

Then, any Monte Carlo method can be applied to solve the equations. We
generate 1500 gluons, 250 up-quarks, 250 down-quarks, 250 up-antiquarks and
250 down-antiquarks within $-0.3<z<0.3$ fm in the longitudinal direction and
$r<6.4$ fm in the transverse direction from HIJING for central Au-Au
collisions at $\sqrt{s_{NN}}=200$ GeV. The number of partons moving in the
longitudinal direction is much larger than the number of partons moving in the
transverse direction. The anisotropically distributed partons undergo elastic
2-to-2 scattering [14, 15] and elastic 3-to-3 scattering to eventually become
isotropic in momentum space (i.e. become a thermal state) at a time. But from
gluon matter to quark matter the time is different, i.e. the thermalization
time of gluon matter differs from the thermalization time of quark matter. At
$t=0.52$ fm/$c$ the gluon momentum distribution meets the Jüttner distribution

$f_{g}(\vec{p})=\frac{\lambda_{g}}{{\rm e}^{\mid\vec{p}\mid/T}-\lambda_{g}},$
(3)

with the temperature $T=0.52$ GeV and the gluon fugacity $\lambda_{g}=0.328$
that label the thermal state of gluon matter. The corresponding thermalization
time is 0.32 fm/$c$. At $t=0.86$ fm/$c$ the quark momentum distribution meets
the Jüttner distribution

$f_{q}(\vec{p})=\frac{\lambda_{q}}{{\rm e}^{\mid\vec{p}\mid/T}+\lambda_{q}},$
(4)

with $T=0.46$ GeV and the quark fugacity $\lambda_{q}=0.143$ that label the
thermal state of quark matter. The corresponding thermalization time is 0.66
fm/$c$. The thermalization time of quark matter is larger than the one of
gluon matter.

##  4 Conclusions

The elastic 2-to-2 scattering and the elastic 3-to-3 scattering lead to the
rapid creation of temperature or the early thermalization of quark-gluon
matter. Solutions of the transport equations offer the thermalization times
and the thermal states of gluon matter and quark matter. Gluon matter
thermalizes faster than quark matter.

This work was supported by the National Natural Science Foundation of China
under grant no 11175111.

## References

## References

  * [1] Xu X-M 2006 Eur. Phys. J. A 29 39
  * [2] Ackermann K H et al (STAR Collaboration) 2001 Phys. Rev. Lett. 86 402
  * [3] Kolb P F, Huovinen P, Heinz U and Heiselberg H 2001 Phys. Lett. B 500 232
  * [4] Teaney D, Lauret J and Shuryak E V 2001 A hydrodynamic description of heavy ion collisions at the SPS and RHIC Preprint nucl-th/0110037
  * [5] Shuryak E 1992 Phys. Rev. Lett. 68 3270
  * [6] Wong S M H 1996 Phys. Rev. C 54 2588
  * [7] Shin G R and Müller B 2003 J. Phys. G: Nucl. Part. Phys. 29 2485
  * [8] Blaizot J-P, Gelis F, Liao J, McLerran L and Venugopalan R 2012 Thermalization and Bose-Einstein condensation in overpopulated plasmas Preprint arXiv:1210.6838
  * [9] Xu X-M, Sun Y, Chen A-Q and Zheng L 2004 Nucl. Phys. A 744 347
  * [10] Xu X-M, Peng R and Weber H J 2005 Phys. Lett. B 629 68
  * [11] Xu X-M, Shen Z-Y, Ye Z-C and Xu W-J 2013 Phys. Rev. C in press
  * [12] Xu X-M, Ma C-C, Chen A-Q and Weber H J 2007 Phys. Lett. B 645 146
  * [13] Xu X-M and Xu L-S 2010 J. Phys. G: Nucl. Part. Phys. 37 115003
  * [14] Cutler R and Sivers D 1978 Phys. Rev. D 17 196
  * [15] Combridge B L, Kripfganz J and Ranft J 1977 Phys. Lett. B 70 234


# Around Homology Planes: Old and new

Rodolfo Aguilar Aguilar  University of Miami, Coral Gables, USA
<EMAIL_ADDRESS>https://sites.google.com/view/rodolfo-aguilar/

###### Abstract.

We survey some old and new results about acyclic (affine) complex surfaces,
also called homology planes. We ask several questions and leave open
directions for future research.

###### Contents

  1. 1 Introduction
  2. 2 Context
    1. 2.1 Cancellation property
    2. 2.2 Classification
  3. 3 Exotic structures
    1. 3.1 Algebraic
    2. 3.2 Analytic
    3. 3.3 Symplectic
    4. 3.4 Contractible four-manifolds
  4. 4 More symplectic properties
    1. 4.1 Search for more exotic symplectic structures
    2. 4.2 A question for the contact structure of the boundary
    3. 4.3 Invariance of log-Kodaira dimension
    4. 4.4 Mirror symmetry
  5. 5 Topology
    1. 5.1 Fundamental groups
    2. 5.2 Boundary manifolds
    3. 5.3 Character varieties
  6. 6 Algebra
    1. 6.1 Higher dimensions
    2. 6.2 Cancellation problem

##  1\. Introduction

This note is to extend a talk given in Sunny Beach on August 2023 during the
conference “Generalized and Symplectic Geometry”. The talk was intended to
give a survey around the classification of acyclic affine complex algebraic
surfaces and to highlight some connections with topology and symplectic
geometry.

Most of the note is expository although there are some new results. These new
results are in the topology section and they will be explained in more detail
below.

In writing these notes, I use the opportunity to present some open questions
that arose in discussions with people in the conference and outside. There are
already some very good surveys about homology planes, see [Zai98] and [Miy01].
For connections with $\mathbb{A}_{1}$-homotopy theory see also [AOsr21]. I try
by no means to be exhaustive, and the contents here only reflects the interest
of the author. He apologizes if he forgot to mention or give some credits.

The paper is organized as follows: in Section 2, we present the cancellation
problem as a motivation for studying homology planes and present some
classification results.

In Section 3, we present some algebraic and analytic cancellation Theorems and
show how this give exotic algebraic and analytic structures on
$\mathbb{C}^{n}$ for $n\geq 3$. After this, we mention how we can obtain
exotic symplectic structures on some affine real spaces. We end the section by
presenting a result about exotic balls.

In Section 4, more questions and symplectic techniques are presented.

In Section 5, we study the relation of the fundamental group of some homology
planes and the fundamental group of the $3$-manifold that can be thought as
its boundary. We present a new result about the structure of this variety in
Theorem 5.7 and a partial computation of its character variety in Theorem 5.9.

Finally, in Section 6, we survey some results and questions in higher
dimension and we finish presenting some open question about the cancellation
problem.

### Acknowledgements

The author is grateful to F. Campana, J. Fernández de Bobadilla, L. Katzarkov,
M. McLean, A. Némethi, D. Pomerleano, P. Popescu-Pampu, N. Saveliev, J.
Svoboda, T. Syed and M. Zaidenberg for interesting discussions around the
topics treated here. He also wants to thank the organizers and participants of
the conference “Generalized and Symplectic Geometry”.

He is thankful to A. Dubouloz, M. Golla, T. Pełka, O. Şavk, for reading and
commenting preeliminaries versions of this note.

##  2\. Context

###  2.1. Cancellation property

Consider smooth affine varieties $X,Y,Z$. We say that the triple111In this
note, we will be mainly interested in the case when $Z=\mathbb{C}^{n}$ for
some $n\geq 1$ and either $X$ or $Y$ will belong to some special class of
quasi-projective varieties, for example, $Y$ is of log-general type. In this
case, we don’t fix the triple but only the special class of varieties, see
Section 3. $(X,Y,Z)$ satisfy the _cancellation property_ if, whenever we have
an isomorphism $X\times Z\cong Y\times Z$ in the category of algebraic
varieties, it induces a commutative diagram:

${X\times Z}$${Y\times Z}$${X}$${Y}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

As stated in [Ram71], it was believed around 1970, that every smooth
contractible (affine), rational surface was isomorphic to $\mathbb{C}^{2}$.

This belief, together with the motivation of the cancellation problem, lead
C.P. Ramanujam to find the following example.

######  Example 2.1 ([Ram71]).

Consider a plane conic $C_{1}$ and a cuspidal cubic $C_{2}$ intersecting as
$C_{1}\cdot C_{2}=P+5Q$. Let $\bar{X}=\operatorname{Bl}_{P}\mathbb{P}^{2}$ and
denote the strict transforms of $C_{1}$ and $C_{2}$ by
$C_{1}^{\prime},C_{2}^{\prime}$ respectively. Let $X=\bar{X}\setminus
C_{1}^{\prime}\cup C_{2}^{\prime}$. C. P. Ramanujam showed that $X$ is affine
contractible and $X\not\cong\mathbb{C}^{2}$.

Figure 1. Dual graph of the minimal embedded resolution of $C_{1}^{\prime}\cup
C_{2}^{\prime}$

Moreover, using the methods for computing the fundamental group developed in
[Mum61], he found a nice topological characterization of the complex affine
plane. Before stating it, let us introduce a definition.

If $X$ is a smooth affine surface, we can consider a compactification
$\bar{X}$ with $D:=\bar{X}\setminus X$ a SNC-divisor. Take a closed regular
tubular neighborhood $U$ of $D$ in $\bar{X}$. The _fundamental group at
infinity_ is $\pi_{1}(\partial U)$.

######  Theorem 2.1 ([Ram71]).

Let $X$ be a smooth affine surface which is contractible and simply connected
at infinity. Then $X$ is isomorphic to $\mathbb{C}^{2}$ as an algebraic
variety.

Ramanujam left open the following question. We will review the solution in
Section 3.1.

######  Question 1 ([Ram71]).

Is $\mathbb{C}^{3}\cong X\times\mathbb{C}$ as an algebraic variety?

###  2.2. Classification

######  Definition 2.1.

Let $X$ be a smooth surface such that $H_{i}(X,\mathbb{Z})=0$ for $i>0$. We
call $X$ a homology plane. If $H_{i}(X,\mathbb{Q})=0$ for $i>0$, we call $X$ a
$\mathbb{Q}$-homology plane.

######  Example 2.2.

The Ramanujam’s surface $X$ is a homology plane.

Using the classification of smooth projective surfaces, the Miyaoka inequality
[Miy84] together with some technical computations, Gurjar and Shastri proved
the following:

######  Theorem 2.2 ([GS89a],[GS89b]).

Homology planes are rational.

The classification of homology planes follows the lines of the classification
of surfaces via the (logarithmic) Kodaira dimension, that we introduce next.

######  Definition 2.2.

Let $X$ be a smooth quasi-projective surface and let $(\bar{X},D)$ be a SNC-
compactification. We define the _logarithmic Kodaira dimension_
$\bar{\kappa}(X)\in\\{-\infty,0,1,2\\}$ as the unique value that satisfies
that, for some positive constants $\alpha,\beta$ we have

$\alpha m^{\bar{\kappa}(X)}\leq\dim H^{0}(\bar{X},m(K_{\bar{X}}+D))\leq\beta
m^{\bar{\kappa}(X)}$

where $m$ is a sufficiently large and divisible positive integer.

The invariant $\bar{\kappa}(X)$ does not depend on the chosen compactification
of $X$. See [Iit77].

When $\bar{\kappa}(X)\leq 1$, we have an easy description for the homology
plane $X$. Recall that a quasi-pencil $L(1,n+1)$ is a set of lines in
$\mathbb{P}^{2}$ having $n$-lines passing through a point and another line in
general position. See Figure 2.

######  Theorem 2.3 ([Fuj82], [GM88], [tDP90]).

Let $X$ be a homology plane.

  1. (1)

If $\bar{\kappa}(X)=-\infty$ then $X\cong\mathbb{C}^{2}$.

  2. (2)

There are no homology planes with $\bar{\kappa}(X)=0$.

  3. (3)

All homology planes with $\bar{\kappa}(X)=1$ arise from a quasi-pencil of
lines $L(1,n)$.

$L_{n}$$L_{n-1}$$L_{2}$$L_{1}$$L_{n+1}$ Figure 2. $L(1,5)$ in the projective
plane

For a homology plane $X$ with $\bar{\kappa}(X)=1$, its construction depends of
certain integer pairs $(a_{1},b_{1}),\ldots,(a_{n},b_{n})$ satisfying

(1) $a_{1}\cdots a_{n}+\sum_{i=1}^{n}a_{1}\cdots b_{i}\cdots a_{n}=\pm 1.$

It is obtained from $L(1,n)$ by blowing-up over some singular points and
removing some $(-1)$-curves. For its precise construction, see [Miy01] or
[AAŞ22, Section 2.2].

Now, when $\bar{\kappa}(X)=2$ we say that $X$ is of log-general type.
Amazingly, due to the constrains imposed by $X$ being a homology plane, there
is a complete (conjectural) classification.

Let us begin first with some conjectures.

######  Conjecture 2.1.

Let $X$ be a homology plane of log general type and let $(\bar{X},D)$ be a
minimal SNC completion of $X$. Then

  1. (1)

(tom Dieck-Petrie conjecture) Every homology plane arises from finitely many
configurations of lines and conics in $\mathbb{P}^{2}$.

  2. (2)

(Koras conjecture) The automorphism group of $X$ has at most six elements,

  3. (3)

(Flenner-Zaidenberg Rigidity conjecture) $H^{i}(\mathcal{T}_{\bar{X}}(-\log
D))=0$ for all $i\geq 0$,

  4. (4)

(Palka negativity conjecture) $\kappa(K_{\bar{X}}+\frac{1}{2}D)=-\infty$.

In [Peł23], T. Pełka completely classified the $\mathbb{Q}$-homology planes
assuming Palka negativity conjecture. He uses the theory of almost minimal
model of Palka [Pal19]. Indeed, assuming that
$\kappa(K_{\bar{X}}+\frac{1}{2}D)=-\infty$, one obtains that a minimal model
$(\bar{X}_{\text{min}},\frac{1}{2}D_{\text{min}})$ is a log-Mori fiber space
over either a curve or a point. The first case was studied by [MS91], although
Pełka gives a self-contained proof. The second case is the main part of his
work. One can then state his main theorem (we will state it only for
$\mathbb{Z}$-homology planes).

######  Theorem 2.4 ([Peł23]).

Assume (4) in Conjecture 2.1 (Palka negativity conjecture), then it implies
(1),(2) and (3). In particular, the $\mathbb{Z}$-homology planes of log-
general type can be arranged in finitely many discrete series, each obtained
in a uniform way from an arrangement of lines and conics on $\mathbb{P}^{2}$.

##  3\. Exotic structures

###  3.1. Algebraic

Using the theory developed in [Iit77], Iitaka-Fujita proved the following
cancellation theorem.

######  Theorem 3.1 ([IF77]).

If we have an algebraic isomorphism $X\times\mathbb{C}^{n}\cong
Y\times\mathbb{C}^{n}$ and $\bar{\kappa}(Y)\geq 0$ then the cancellation
property holds: $X\cong Y$.

By taking $Y$ to be the Ramanujam surface of Example 2.1 (it can be shown that
$\bar{\kappa}(Y)=2$), we can obtain algebraic exotic structures on
$\mathbb{C}^{n}$ for $n\geq 3$.

######  Corollary 1.

The space $\mathbb{C}^{n}$ admits an exotic algebraic structure for $n\geq 3$.

Indeed, using the $h$-cobordism Theorem one can show that
$Y\times\mathbb{C}^{n}$ is diffeomorphic to $\mathbb{R}^{2n+4}$. By taking
$X=\mathbb{C}^{2}$ and if we suppose that $X\times\mathbb{C}^{n}\cong
Y\times\mathbb{C}^{n}$, we use the cancellation Theorem 3.1 of Iitaka-Fujita,
which would give the contradiction that the Ramanujam surface is isomorphic to
$\mathbb{C}^{2}$.

###  3.2. Analytic

Using properties of non-singular quasi-projective varieties of log-general
type (e.g. every dominant holomorphic map $X\to Y$ from a quasi-projective
manifold to a manifold of log-general type is regular), Zaidenberg proved the
following _analytic_ cancellation theorem.

######  Theorem 3.2 ([Zai93]).

Suppose that $X$ and $Y$ are smooth surfaces of log-general type, this is
$\bar{\kappa}(X)=\bar{\kappa}(Y)=2$, then $X,Y,\mathbb{C}^{n}$ satisfy an
_analytic_ cancellation property.

Moreover, he constructed two infinite families of contractible homology planes
of log-general type pairwise non-isomorphic. One family generalized the
Ramanujam surface.

Using these infinite families, his analytic cancellation theorem 3.2 and the
same argument in the proof of Corollary 1, he obtained the following
Corollary.

######  Corollary 2 ([Zai93]).

There is a countable infinite set of analytic exotic structures on
$\mathbb{C}^{n}$ for $n\geq 3$.

Moreover, in [FZ94], by using that homology planes of log-Kodaira dimension
one can have moduli, they obtained an _uncountable_ number of exotic
structures on $\mathbb{C}^{n}$ for $n\geq 3$.

###  3.3. Symplectic

In [SS05], the authors studied the symplectic topology of the Ramanujam
surface. More precisely, they obtained the following.

######  Theorem 3.3 ([SS05]).

Let $X$ be the Ramanujam’s surface, the product $X^{m}\cong\mathbb{R}^{4m}$
induces an exotic symplectic structure on $\mathbb{R}^{4m}$ which is convex at
infinity.

As explained in [Sei08], this can be shown by showing that the symplectic
cohomology of $X^{m}$ does not vanish.

To construct an infinite countable number of exotic symplectic structures on
$\mathbb{R}^{2m}$ for $m>3$ McLean started from a different example. He used
the so-called Kaliman modification [Kal94] to construct Lefschetz fibrations.
As the cancellation theorem are no longer valid in the symplectic category, he
used the number of idempotents in symplectic homology as invariant to
distinguish between different exotic structures.

######  Theorem 3.4 ([McL09]).

Let $m>3$. Then for each $\mathbb{R}^{2m}$ there are infinitely many finite
type Stein manifolds which are pairwise distinct as symplectic manifolds.

######  Question 2.

Is it possible to construct uncountably many exotic symplectic structures on
$\mathbb{R}^{2m}$ for $m>3$?

Note that one needs a stronger invariant than symplectic cohomology if one
wants to use the deformation examples of Flenner-Zaidenberg.

###  3.4. Contractible four-manifolds

An important problem in low-dimensional topology is to determine which
homology $3$-spheres bound contractible $4$-manifolds. See [Şav24].

A _Mazur_ manifold, is a contractible, compact, smooth $4$-dimensional
manifold with boundary, which is not diffeomorphic to the standard $4$-ball
and such that in its handle decomposition it contains exactly three handles:
one $0$-handle, one $1$-handle and one $2$-handle.

Let us review briefly the construction of Mazur and Poénaru manifolds as in
[Şav20, Sec. 2.2] which is closer to the construction by V. Poénaru [Poe60].
Consider a disk $D$ smoothly embedded in $B^{4}$ and take a tubular
neighborhood $U$ of $D$. In the boundary $\partial U$, which is diffeomorphic
to $S^{1}\times S^{2}$, make an integral surgery in a knot $J$ with fusion
number $0$ such that the $3$-manifold obtained is a homology sphere. The
resulting $3$-manifold bounds a Mazur manifold, see [Şav20, Lemma 2.1].

In this construction, if we replace the disk $D$ by a ribbon disk $D\subset
B^{4}$ and the knot $J$ by a ribbon knot with fusion number $n\geq 1$, the
resulting $3$-manifold bounds a contractible four manifold with one
$0$-handle, $n+1$ $1$-handles and $n+1$ $2$-handles. This is called a _Poénaru
manifold_.

In order to obtain this ribbon knot, one fixes a compactification
$(\bar{X},D)$ of a homology plane $X=\bar{X}\setminus D$ and consider the dual
graph of $D$. This can also be seen as the surgery diagram of the boundary of
a tubular neighborhood of $D$ as follows: to every vertex associate an unknot,
link two unknots if and only if there is an edge connecting the corresponding
vertices and decorate the unknots with the weights of the corresponding
vertices. If by applying Kirby calculus one ends with a ribbon knot, one uses
Şavk’s construction of Poénaru manifolds.

By using the vast quantity of example in the literature of homology planes, in
particular some examples of Zaidenberg generalizing the Ramanujam surface and
contractible homology planes of log-Kodaira dimension one and using Kirby
calculus, one can show the following Theorem.

######  Theorem 3.5 ([AAŞ22]).

There are four infinite families of homology $3$-spheres that bound both
homology planes and Mazur/Poénaru manifolds.

In [AAŞ22], these homology $3$-spheres bounding both homology planes and
Mazur/Poénaru manifolds are called Kirby-Ramanujam spheres. In Theorem 3.5,
one infinite family consists of Mazur manifolds and the other three of Poénaru
manifolds.

##  4\. More symplectic properties

###  4.1. Search for more exotic symplectic structures

One can ask if by using other examples of contractible homology planes one can
obtain more exotic symplectic structures on $\mathbb{R}^{4m}$. This is the
content of the unpublished work of D. Jackson-Hanen [JH14].

Moreover, partially motivated by this kind of quest, Ganatra-Pomerleano showed
the following Theorem.

######  Theorem 4.1 ([GP20]).

Let $X$ be a smooth affine variety and let $(\bar{X},D)$ be a SNC-
compactification. Then there is a multiplicative spectral sequence converging
to the symplectic cohomology ring

$\\{E_{r}^{p,q},d_{r}\\}\to SH^{*}(X)$

whose first page is isomorphic as rings to the logarithmic cohomology of
$(\bar{X},D)$.

For the definition of logarithmic cohomology of a pair $(\bar{X},D)$, see
[GP21].

######  Question 3.

  1. (1)

Can one use this spectral sequence to recover the Seidel-Smith result?

  2. (2)

Can one use the classification of Pełka to obtain more exotic structures?

###  4.2. A question for the contact structure of the boundary

If $X=\bar{X}\setminus D$ is a homology plane and we let $U$ be a closed
regular tubular neighborhood of $D$, then the boundary of $\bar{X}\setminus U$
inherits a contact structure.

We can say that an oriented contact $3$-manifold $(M,\xi)$ is Ramanujam
fillable if it is contactomorphic to the contact boundary of a homology plane.

We can try to extend the following result of singularity theory.

######  Definition 4.1.

Let $(M,\xi)$ be an oriented contact manifold. We say it is Milnor fillable if
it is contactomorphic to the contact boundary of an isolated complex-analytic
singularity $(\mathcal{X},x)$.

######  Theorem 4.2 ([CNPP06]).

Let $(M,\xi)$ be any three dimensional oriented manifold. Then it admits at
most one Milnor fillable structure up to contactomorphism.

######  Question 4.

Does an analog of the Caubel, Némethi, Popescu-Pampu theorem 4.2 holds in this
setting (probably restricting the Kodaira dimension)?

In fact, as shown in [Peł23, Cor 1.10], there exists examples of non
(algebraically) isomorphic but diffeomorphic homology planes
$X=\bar{X}\setminus D$ with the same dual graph for $D$. These are sometimes
called Zariski-pairs, see [ABCT08]. This indicates that additional information
to the dual graph is needed, we ask if the contact structure is enough for
this. Moreover, similar questions and techniques have been studied in the
symplectic category for rational cuspidal curves. See [GS22, GK23].

###  4.3. Invariance of log-Kodaira dimension

Let us gather here more questions related to homology planes.

In the compact setting, the invariance of the Kodaira dimension under
diffeomorphism is shown by Friedman-Morgan [FM94]. In the symplectic category
we have the following result of McLean.

######  Theorem 4.3 ([McL14]).

The logarithmic Kodaira dimension is a symplectic invariant of homology
planes.

To prove this, he uses Theorem 2.1, 2.3 and the theory he develops about the
symplectic invariance of uniruledness.

######  Question 5.

(McLean): Is the logarithmic Kodaira dimension a topological invariant of
homology planes?

###  4.4. Mirror symmetry

When $X$ is a contractible homology plane of log-Kodaira dimension one, it is
shown in [tDP90] that they can arise as hypersurfaces in $\mathbb{C}^{3}$. It
could be interesting to compute their mirror using the techniques in [AAK16].

Moreover, as they are contractible, one can wonder what kind of invariants
could be non-trivial in the mirror. Recall that for example, in the Ramanujam
surface, symplectic cohomology is non-trivial.

The following question is due to D. Pomerleano.

######  Question 6.

Is it possible to use the homology planes to construct exotic
$(\mathbb{C}^{*})^{r}$ structures?

Note that, if one wants to follow the proof of Corollary 1, one would need to
use the s-cobordism Theorem, instead of the h-cobordism.

##  5\. Topology

###  5.1. Fundamental groups

Motivated by the question, often attributed to Serre, of what groups can
appear as fundamental groups of complex projective manifolds, we can ask:

  1. (1)

What groups can appear as fundamental groups of homology planes,

  2. (2)

What groups can appear as fundamental groups at infinity for homology planes,

  3. (3)

What is the relation between these two?

A first constrain in given by the following theorem of S. Orevkov.

######  Theorem 5.1 ([Ore97]).

Let $X=\bar{X}\setminus D$ be a homology plane and for $D\subset U$ a tubular
neighborhood as above then the three-manifold $\partial U$ is not a Seifert
fibered homology sphere.

An _orbicurve group_ is a group admitting a presentation of the following
type:

$\displaystyle\langle a_{1},b_{1},\ldots,a_{g},b_{g},c_{1},\ldots,c_{s}\mid$
$\displaystyle\quad\quad
c_{1}^{r_{1}}=\ldots=c_{s}^{r_{s}}=1,\prod_{i=1}^{g}[a_{i},b_{i}]c_{1}\ldots
c_{r}=1\rangle$

######  Theorem 5.2 ([AA21]).

There exists infinitely many homology planes of log-general type with infinite
fundamental group. All of them admit a quotient to an infinite orbicurve
group.

For the proof, a presentation of the fundamental group of the homology plane
is obtained. Then, some simpler quotient (an orbicuve group) which is known to
present an infinite group is deduced.

######  Question 7.

Does every infinite fundamental group of a homology plane admits a surjective
morphism to an infinite orbicurve group?

By a Theorem of Arapura [Ara97], extended to orbicurves in [ABCAM13], the
morphisms from quasi-projective varieties to orbicurves are encoded in the
irreducible components of the so-called cohomology jump loci. For a quasi-
projective variety $X$, this loci consists of elements $\rho$ in
$H^{1}(X,\mathbb{C}^{\ast})$ such that, the associated local system $V_{\rho}$
has non-vanishing first cohomology $H^{1}(X,V_{\rho})\not=0$.

Now, from the construction of homology planes coming from an arrangement of
lines $\mathcal{A}$, every point of multiplicity three of higher will give a
map to $\mathbb{P}^{1}$ with some orbifold structure. However, to find the
examples in Theorem 5.2, we used the complete description of the cohomology
jump loci as given by Suciu in [Suc01]. In particular, we used maps not coming
from points of multiplicity three or higher, but others reflecting the global
structure of $\mathcal{A}$ plus some carefully chosen numerical data that
depend on the homology plane.

We do not know if the question 7 has a positive answer even if we assume the
negativity conjecture 2.1, but for homology planes of Kodaira dimension one it
has indeed a positive answer. See below remark 5.5.

If one would like to study the cohomology jump loci in the homology plane
directly and not in the complement of an arrangement $\mathcal{A}$ in
$\mathbb{P}^{2}$ a generalization of Arapura’s result is needed for non-
abelian representations, for example in $SL_{2}(\mathbb{C})$. Even simpler, we
can try to study the $SU(2)$ character varieties. This is not easy, even for
the representations of the $3$-homology sphere bounding a homology plane $X$,
as we will see in subsection 5.3. For more of representations of quasi-
projective groups on $SL_{2}(\mathbb{C})$, see [CS08, PS20].

When $X$ is a homology plane of log-Kodaira dimension one. It is easy to
compute its fundamental group.

######  Proposition 5.3.

Let $X$ be a homology plane of logarithmic Kodaira dimension one coming from
$L(1,n+1)$ with data $(a_{1},b_{1}),\ldots,(a_{n},b_{n})$ then a presentation
for its fundamental group is given by

(2)
$\langle\gamma_{0},\ldots\gamma_{n}\mid\gamma_{1}\cdots\gamma_{n}=\gamma_{0}^{-1},[\gamma_{0},\gamma_{i}]=1,\gamma_{i}^{a_{i}}\gamma_{0}^{b_{i}}=1\text{
for }i=1,\ldots,n\rangle$

######  Remark 5.4.

Note that from (2), we obtain that $H_{1}(X)$ is trivial if and only if

$\det\left(\begin{array}[]{rrrr}a_{1}-b_{1}&-b_{1}&\ldots&-b_{1}\\\
-b_{2}&a_{2}-b_{2}&&-b_{2}\\\ \vdots&&\ddots&\\\
-b_{n}&-b_{n}&&a_{n}-b_{n}\par\end{array}\right)=\pm 1$

which is precisely the condition in (1).

######  Remark 5.5.

Note that all homology planes obtained from $L(1,3)$ are contractible. Indeed,
note that from (2), more specifically, from $[\gamma_{0},\gamma_{2}]=1$ and
$\gamma_{0}^{-1}=\gamma_{1}\gamma_{2}$ we obtain that
$[\gamma_{1},\gamma_{2}]=1$ and hence $\pi_{1}(X)$ is abelian. By the
condition (1) and by Remark (5.4), we obtain that $\pi_{1}(X)$ is trivial.

Moreover, when $n\geq 3$, the homology planes obtained from $L(1,n+1)$ by the
above procedure are never contractible. Indeed, it admits an orbifold morphism
to $\mathbb{P}^{1}$ with $n$ marked points with weights $a_{1},\ldots,a_{n}$.

######  Remark 5.6.

Note that the presentation in (2) is the presentation of the fundamental group
of a Seifert fibered homology sphere.

###  5.2. Boundary manifolds

When $X=\bar{X}\setminus D$ is a homology plane of log-Kodaira dimension one,
the dual graph of $D$ is as in Figure 3 or it can have an extra linear segment
of $-2$-curves corresponding to a series of blow-ups starting in a smooth
point in the line $L_{n+1}$.

Recall Theorem 3.5. In [AAŞ22], for an infinite family $(\bar{X}_{n},D_{n})$
of homology planes of logarithmic Kodaira dimension one, it was shown that if
we let $\partial U_{n}$ be the boundary of a regular tubular neighborhood of
$D_{n}$, then it is the splice of two Seifert fibered spheres
$\Sigma(a_{1n},a_{2n},a_{1n}a_{2n}\pm 1)$ glued along the last fiber. That is,
in each Seifert fibered homology sphere $\Sigma(a_{1n},a_{2n},a_{1n}a_{2n}\pm
1)$ a small neighborhood of the third twisted fiber is removed, and the
boundaries are glued longitude-to-meridian.

Using the techniques in [NR78]. Together with A. Némethi we obtained the
following Theorem.

######  Theorem 5.7 (A-Némethi ’22).

Let $X$ be a homology plane with $\bar{\kappa}(X)=1$ and suppose that the dual
graph of $D$ is as in Figure 3. Let $\partial U$ be as above. Then the
homology sphe $\partial U$ is the splice of

$\Sigma(a_{1},\ldots,a_{n},a_{1}\cdots a_{n}\pm 1)$

along the last fiber.

It is easy to compute the fundamental group of $\partial U$ by using the
methods of [Mum61]. Moreover, by fixing explicit geometric meridians around
the lines in $L(1,n)$, it is possible to compute the presentation in (2) as a
quotient from a presentation of $\pi_{1}(\partial U)$. We will see in the next
section that it is however, very hard to compute and find relations between
invariants of these four and three (real) manifolds.

###  5.3. Character varieties

Let $X$ be a compact manifold, the $SU(2)$ _character variety_ of $X$ is
defined to be the topological space

$R(X)=\operatorname{Hom}(\pi_{1}(X),SU(2))/conj$

For homology planes of log-Kodaira dimension, by Theorem 5.7, we have that the
$3$-manifold $\partial U$ the splice of two Seifert spheres

$\Sigma(a_{1},\ldots,a_{n},a_{1}\cdots a_{n}\pm 1)$

along the last fiber.

Figure 3. Splice of the Seifert spheres $\Sigma(a_{1},\ldots,a_{n},a_{1}\cdots
a_{n}\pm 1)$ along its last fiber.

When the splice components have only three singular fibers, the character
variety is studied by N. Saveliev in [Sav98]. Interestingly, it turns out that
it is simply the disjoint union of the two splice component.

######  Theorem 5.8 ([Sav98]).

Let $\Sigma$ be the splice of two Seifert spheres
$\Sigma(a_{1},a_{2},a_{1}a_{2}\pm 1)$ along its last fiber. Then

$R(\Sigma)=R(\Sigma(a_{1},a_{2},a_{1}a_{2}+1))\coprod
R(\Sigma(a_{1},a_{2},a_{1}a_{2}-1))$

Using this, he can conclude that instanton Floer homology groups
$I_{j}(\Sigma)$ vanish for odd $j$.

Now, if we want to generalize this computations to the case when $n>2$, this
is, if the splice components have more than $3$ singular fibers, the
computations are much more complicated and only partial results can be
obtained with similar computations.

Denote $\Sigma_{\pm}=\Sigma(a_{1},\cdots,a_{n},a_{1}\cdots a_{n}\pm 1)$ and
let

$\mathcal{R}(\Sigma_{\pm})=\mathcal{R}_{\pm}\coprod\mathcal{R}(a_{1},\ldots,a_{n})$

where $\mathcal{R}_{+}$ is the union of the components of
$\mathcal{R}(\Sigma_{+})$ consisting of those representations $\rho$ that send
the generator corresponding to the fiber last fiber $a_{1}\ldots a_{n}+1$ to a
non-trivial element. A similar description holds for $\mathcal{R}_{-}$.

The following description was obtained in discussions with N. Saveliev. The
proof is analogous to that of Theorem 5.8.

######  Theorem 5.9 (A-Saveliev ’23).

Let $\Sigma$ be the splicing of $\Sigma_{+}$ and $\Sigma_{-}$ along the $n+1$
singular fiber. Then

$\mathcal{R}(\Sigma)\supset
E\coprod\left(\mathcal{R}_{+}\coprod\mathcal{R}_{-}\right),$

where $E$ is an $SO(3)$-fiber bundle over
$\mathcal{R}(a_{1},\ldots,a_{n})\times\mathcal{R}(a_{1},\ldots,a_{n})$.

The difficult part comes from those representations of $\pi_{1}(\Sigma)$ that
restrict to a central representation in one component and are irreducible in
the other. Of course, this also complicates the computation of instanton Floer
homology groups.

It may happen that other invariants could give more insight in the relation
between the geometric and algebraic structure. For example if the intersection
matrix of $D$ is negative definite, the analytic lattice cohomology of A.
Némethi and collaborators gives nice relations, see [ÁN21]. However, if the
matrix is no longer negative definite, it is difficult to conclude. It could
be interesting if the invariants of [GPPV20] could be computed (if defined)
for $\partial U$.

##  6\. Algebra

###  6.1. Higher dimensions

It is a result from Fujita [Fuj82], that homology planes are affine surfaces.
In particular they are Stein. If we consider contractible algebraic varieties
of higher dimension, Winkelmann showed that this is no longer the case.

######  Theorem 6.1 ([Win90]).

There exists a contractible quasi-affine variety $X$ such that $X$ is non-
Stein.

The example is obtained by the quotient of a free action of the additive group
$\mathbb{G}_{a}$ on $\mathbb{A}^{5}$. One may wonder if more examples of this
type could be constructed.

######  Theorem 6.2 ([ADF17]).

Let $n\geq 3$. There exists contractible varieties $X_{2n}$ of dimension $2n$
which are not a quotient of affine space by a free action of a unipotent
group.

The examples are given by removing a closed subscheme $E$ isomorphic to
$\mathbb{A}^{n}$ from certain affine quadric $Q_{2n}$.

One may also ask of the higher dimensional generalization of the Theorem 2.2
of Gurjar-Shasti holds. This being probably too hard, can be weakened to the
following question of F. Campana.

######  Question 8 (Campana).

Let $X$ be a smooth, contractible affine variety $X$. Consider a smooth
compactification $\bar{X}$ of $X$, is then $\bar{X}$ rationally connected?

Similar questions about compactifications of smooth contractible threefolds
have been treated by T. Kishimoto and collaborators. See [Kis04, Kis05].

The following comment was made by A. Dubouloz.

It is remarkable the existence of exotic algebraic structures on
$\mathbb{C}^{3}$ that are rational, affine, of logarithmic Kodaira dimension
$-\infty$ and such that it is not a product of a contractible surface with
$\mathbb{C}$.

Probably, the most famous example is Russell’s cubic $X$ defined by the
equation ${x^{2}y+z^{2}+t^{3}+x=0}$ in $\mathbb{C}^{4}$. Topologically, it is
a variety diffeomorphic to $\mathbb{R}^{6}$, but algebraically different from
$\mathbb{C}^{3}$. The following open questions seem far from being answered.

######  Question 9.

  1. (1)

Is $X$ analytically isomorphic to $\mathbb{C}^{3}$?

  2. (2)

Is $X\times\mathbb{C}$ algebraically/analytically isomorphic to
$\mathbb{C}^{4}$?

In the symplectic context, it is shown in [CM19] that $X$ is Stein
deformation-equivalent to $\mathbb{C}^{3}$.

###  6.2. Cancellation problem

To finish this note, let us conclude with an unexpected connection , at least
to the author, between the cancellation property and moduli spaces of
representations on a punctured Riemann surface.

Let $M_{\mathcal{B}}$ be the Betti moduli space of stable filtered regular
$G$-local systems over a punctured Riemann surface. Denote the complex
dimension of $M_{\mathcal{B}}$ by $d$. When $M_{\mathcal{B}}$ is very generic
(see Section 1.2 in [Su23]), we have a strong form of Mellit’s cell
decomposition:

######  Theorem 6.3 ([Su23]).

$\mathcal{M}_{B}$ is decomposed into locally closed subvarieties of the form
$(\mathbb{C}^{*})^{d-2b}\times\mathcal{A}$, where $\mathcal{A}$ is stably
isomorphic to $\mathbb{C}^{b}$.

Expectation (Su): $\mathcal{A}$ should give in general a counter-example to
the Zariski cancellation problem for dimension bigger than $3$ in
characteristic zero.

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