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Let $M$ be an oriented closed $3$-manifold equipped with an effective smooth circle action. Can we have a classification of all such $M$ such that there exists a $4$-manifold $N$ with $\partial N=M$, …

A $3$-manifold $M$ is called $P^2$-irreducible if it is irreducible and there is no $2$-sided $P^2$ contained in $M$. Can we show $M$ is $P^2$-irreducible iff $\pi_2(M)=0$? Notice that one direction f …

Let $M$ be a compact $3$-manifold such that no component of $\partial M$ is $S^2$ and one component $F$ of $\partial M$ is the projective plane. If $i_*:\pi_1(F) \to \pi_1(M)$ is an isomorphism, can w …

Let $M^3$ be a compact $3$-manifold such that $\pi_1(M)$ contains a normal subgroup isomorphic to $\mathbb Z$. Can we show either $\pi_1(M)$ is torsion-free or $\pi_1(M)=\mathbb Z \oplus \mathbb Z_2$ …

Let $M^3$ be a closed orientable $3$-manifold. If $H_2(M,\mathbb Z)=0$ and $H_2(M, \mathbb Z_2)\ne 0$, can we show that $M$ contains a 1-sided incompressible surface?