Timeline for Is every codimension-one homology class of a closed manifold represented by a $\pi_1$-injective embedded submanifold?

10 events

when toggle format	what		by	license	comment
May 13 at 12:36	vote	accept	24601
May 12 at 18:29	answer	added	HJRW		timeline score: 5
May 12 at 6:09	comment	added	HJRW		@IanAgol: it follows for general reasons, because the kernel is finitely generated and normal. Maybe I’ll write this up in an answer.
May 11 at 12:53	comment	added	Ian Agol		@HJRW does $ker\{ F \times F \to Z\}$ not split as an amalgamated product over a finitely presented subgroup? If that’s true, then I think your suggestion gives a counterexample. Maybe that follows from the classification of finitely presented subgroups of $F\times F$.
May 8 at 8:37	comment	added	HJRW		A more subtle kind of counterexample occurs in dimension 4. Let $\pi_1(M^4)=\mathbb{Z}^5$ (or, more generally, any $PD_n(\mathbb{Q})$-group for $n>4$). Then any edge group of a splitting must be 4-dimensional by Mayer—Vietoris, but 3-manifold groups are (rationally) 3-dimensional by the sphere theorem.
May 8 at 7:11	comment	added	HJRW		As @IanAgol points out below, the dual cohomology class needs to be dominated by a splitting with finitely presented edge group. The most well known negative example is the map $F\times F\to\mathbb{Z}$ sending each generator to $1$.
May 8 at 0:02	answer	added	Moishe Kohan		timeline score: 5
May 7 at 17:13	comment	added	24601		Apologies, I forgot to add that I also need the submanifold to be compact.
May 7 at 17:12	history	edited	24601	CC BY-SA 4.0	added 12 characters in body
May 7 at 16:24	history	asked	24601	CC BY-SA 4.0