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Realizations of countable groups as fundamental groups of compacta
Virk, Žiga, 1982-
Vprašanje o realizaciji končnih grup v obliki fundamentalnih grup kompaktnih metričnih prostorov je bilo dolgo časa odprto. Take realizacije je razmeroma lahko konstruirati v okviru metričnih ali ...kompaktnih prostorov. Kombinacija obeh lastnosti pa je za fundamentalno grupo zelo restriktivna. Problem so obravnavali mnogi topologi (vključno s Cannonom in Connerjem) vendar do rešitve niso prišli. V tem članku dokažemo, da je možno vsako končno grupo realizirati kot fundamentalno grupo kompaktnega podprostora v ▫${\mathbb R}^4$▫. V skladu z izrekom Shelaha [S. Shelah, Can the Fundamental (Homotopy) Group of a Space be the Rationals?, Proc. Amer. Math. Soc. 103, no. 2, (1988), 627-632] omenjeni prostori niso lokalno s potmi povezani, če grupa ni končno predstavljiva. Izrek je dokazan z eksplicitno konstrukcijo prostora ▫$X_G$▫ za vsako števno grupo ▫$G$▫.
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