The fundamental group of partial compactifications of the complement of a real line arrangement

On homology planes and contractible 4‐manifolds

“…The resulting dual graphs depicted in Figure 9 are called tom Dieck–Petri homology planes . Remark In order to obtain a homology plane from the conditions $$(a,b)$$ above, they must satisfy: $$4b-3a=\pm 1$$, see [2, section 5.2.3].…”

“…Since there are several examples of homology planes that are known to be non‐contractible [2, 38], we curiously ask the following question. Compare with [60, Problem G].…”

“…These meridians can be included in the generators of the boundary of the homology plane $$\pi _1(\partial U)$$. For the explicit description, see [2]. Problem For a non‐contractible homology plane $$X$$, study $$\operatorname{Hom}(\pi _1(X), \operatorname{SU}(2))$$ the representation variety of the Ramanujam sphere $$\partial X$$ as a subvariety of the representation variety $$\operatorname{Hom}(\pi _1(\partial X), \operatorname{SU}(2))$$ of $$X$$.…”