2\mathbb{Z})$

7 events

when toggle format	what		by	license	comment
Aug 24, 2018 at 16:08	vote	accept	john mangual
Aug 2, 2018 at 8:06	history	edited	HJRW	CC BY-SA 4.0	Corrected a mistake imported from Scott's paper.
Jul 31, 2018 at 10:23	comment	added	HJRW		@AllenHatcher: thanks for the useful clarification. I’ll correct the answer shortly.
Jul 30, 2018 at 21:29	comment	added	Allen Hatcher		This seems to be a mistake in Scott's paper. He says that certain subgroups of the unit quaternion group $S^3$ are either cyclic, dihedral, or generalized quaternion (also known as binary dihedral). From this he deduces in particular that dihedral groups act freely on $S^3$. However, $S^3$ has a unique element of order $2$, the quaternion $-1$, so $S^3$ cannot contain dihedral subgroups. On Scott's webpage there is a list of errata for this paper but it does not include this one.
Jul 30, 2018 at 8:58	comment	added	HJRW		@AllenHatcher: Then I must have misunderstood the nomenclature. What should one call the quotients of the 3-sphere by the free dihedral actions that Scott describes on page 451 of his article?
Jul 29, 2018 at 20:37	comment	added	Allen Hatcher		The fundamental groups of prism manifolds are not the dihedral groups themselves but the {\it binary\/} dihedral groups $D^_{2n}$ which are the preimages of the subgroups $D_{2n}\subset SO(3)$ under the projection $S^3\to SO(3)$. Thus $D^_{2n}$ maps onto $D_{2n}$ with a ${\mathbb Z}/2$ kernel.
Jul 29, 2018 at 10:46	history	answered	HJRW	CC BY-SA 4.0