Let $M^3$ be a compact $3$-manifold (possibly with boundary). Suppose $M$ is aspherical, can we show that $M$ must be irreducible? Here, irreducible means any embedded sphere in $M$ bounds a $3$-ball.
$\begingroup$
$\endgroup$
5
-
$\begingroup$ Have you looked at an introductory 3-manifolds notes, like Hatcher or Jaco? $\endgroup$– Ryan BudneyCommented Feb 1, 2021 at 9:52
-
$\begingroup$ @RyanBudney The related result in Hatcher's book is Proposition 3.10, which states that if $\pi_2=0$, any embedded sphere bounds a contractible 3-manifold. This 3-manifold is a 3-ball if we use the result of Poincare's conjecture. I was wonder if there exists a more direct proof for this fact. $\endgroup$– TotoroCommented Feb 1, 2021 at 12:49
-
4$\begingroup$ You absolutely need PC for this, as this statement implies PC. $\endgroup$– Moishe KohanCommented Feb 1, 2021 at 22:54
-
1$\begingroup$ Imagine that the PC fails and $\Sigma$ is a fake 3-dimensional sphere. Take the connected sum $M=T^3 \# \Sigma$. Now, prove that $M$ is aspherical but reducible. $\endgroup$– Moishe KohanCommented Feb 2, 2021 at 11:38
-
$\begingroup$ @MoisheKohan You are right. $\endgroup$– TotoroCommented Feb 2, 2021 at 11:42
Add a comment
|