Timeline for Group of parallelizations of $M^3$ finitely generated?

9 events

when toggle format	what		by	license	comment
Dec 2, 2019 at 16:58	vote	accept	user101010
Nov 4, 2019 at 18:16	answer	added	mme		timeline score: 2
Oct 30, 2019 at 17:54	comment	added	Igor Belegradek		The cover $S^3\to SO(3)$ is principal bundle with fiber $\mathbb Z_2$, so it is classified by maps into $B\mathbb Z_2=RP^\infty$.
Oct 30, 2019 at 17:27	history	edited	user101010	CC BY-SA 4.0	added 11 characters in body
Oct 30, 2019 at 17:27	comment	added	user101010		@IgorBelegradek Oh yes - orientable - thanks. What is the classifying map for 2-fold covers?
Oct 30, 2019 at 16:05	comment	added	Igor Belegradek		For $TM$ to be trivial $M$ must be orientable. The classifying map for the 2-fold covering $S^3\to SO(3)$ induces an exact sequence of pointed sets $[M, S^3]\to [M, SO(3)]\to [M, RP^\infty]$. If $M$ is closed oriented, the degree identifies $[M, S^3]$ and $H^3(M)$, and also $[M, RP^\infty]=H^1(M;\mathbb Z_2)$ which are finitely generated abelian groups. It remains to see if the maps in the exact sequence are group homomorphisms.
Oct 30, 2019 at 11:00	comment	added	YCor		The last sentence would better fit as a separate question.
Oct 30, 2019 at 10:57	history	edited	YCor		edited tags
Oct 30, 2019 at 9:49	history	asked	user101010	CC BY-SA 4.0