Editing Conjecture (section)

===Hauptvermutung===
{{main|Hauptvermutung}}

The [[Hauptvermutung]] (German for main conjecture) of [[geometric topology]] is the conjecture that any two [[Triangulation (topology)|triangulations]] of a [[triangulable space]] have a common refinement, a single triangulation that is a subdivision of both of them. It was originally formulated in 1908, by [[Ernst Steinitz|Steinitz]] and [[Heinrich Tietze|Tietze]].<ref>{{Cite web|url=https://www.maths.ed.ac.uk/~v1ranick/haupt/|title=Triangulation and the Hauptvermutung|website=www.maths.ed.ac.uk|access-date=2019-11-12}}</ref>

This conjecture is now known to be false. The non-manifold version was disproved by [[John Milnor]]<ref>{{Cite journal|first=John W.|last= Milnor |title=Two complexes which are homeomorphic but combinatorially distinct|journal= [[Annals of Mathematics]]|volume=74|year=1961|issue= 2 |pages=575&ndash;590|mr=133127|doi=10.2307/1970299|jstor=1970299}}</ref> in 1961 using [[Analytic torsion|Reidemeister torsion]].

The [[manifold]] version is true in [[dimension]]s {{nowrap|1=''m'' ≤ 3}}. The cases {{nowrap|1=''m'' = 2 and 3}} were proved by [[Tibor Radó]] and [[Edwin E. Moise]]<ref>{{cite book | last = Moise | first = Edwin E. | title = Geometric Topology in Dimensions 2 and 3 | publisher = New York : Springer-Verlag | location = New York | year = 1977 | isbn = 978-0-387-90220-3 }}</ref> in the 1920s and 1950s, respectively.