On T. Petrie’s problem concerning homology planes

Smooth Q$\mathbb {Q}$‐homology planes satisfying the negativity conjecture

“…Some generalizations of Ramanujam's construction are given in [49]. In [49, appendix], Miyanishi and Sugie constructed a series of $$\mathbb {Z}$$‐homology planes of log general type.…”

“…Some generalizations of Ramanujam's construction are given in [49]. In [49, appendix], Miyanishi and Sugie constructed a series of $$\mathbb {Z}$$‐homology planes of log general type. We recover them in $$\langle \EUmathfrak{36}\rangle$$ by expansions with centers $$(L_{1},L_{2})$$, $$(E_{q_{1}},L_{q_{1}q_{2}})$$, $$(L_{1}^{\prime },L_{1})$$, $$(L_{2}^{\prime },L_{2})$$ and weights $$(m^{-1},n-2,r+1,(n-m)^{-1})$$, where $$r=m(n-1)$$ or $$m(n-1)-2$$.…”

“…We recover them in $$\langle \EUmathfrak{36}\rangle$$ by expansions with centers $$(L_{1},L_{2})$$, $$(E_{q_{1}},L_{q_{1}q_{2}})$$, $$(L_{1}^{\prime },L_{1})$$, $$(L_{2}^{\prime },L_{2})$$ and weights $$(m^{-1},n-2,r+1,(n-m)^{-1})$$, where $$r=m(n-1)$$ or $$m(n-1)-2$$. By [49, Theorem 2], the surfaces with $$r=m(n-1)$$ are contractible.…”

Smooth $\mathbb{Q}$-homology planes satisfying the Negativity Conjecture

“…[tDP93,3.15]. Some generalizations of Ramanujam's construction are given in [Sug90]. In the Appendix to loc.…”