2
votes
1answer
143 views

isotopy classes of embeddings of the torus

Let's consider $S^1$-bundle $E$ over a 2-manifold $M$. How many isotopy classes of embeddings of the torus $\mathbb{T}^2$ in $E$? For each free homotopy classes $\gamma$ of mappings of the circle ...
4
votes
3answers
363 views

Second Homotopy Group of Graph Manifolds

A graph manifold is a closed 3-manifold $M$ that admits a finite collection of disjoint embedded tori $\mathcal{T}$ so that $M \setminus \mathcal{T}$ is a disjoint union of Seifert fibred spaces (i.e. ...
4
votes
2answers
327 views

Does a homeomorphism of $S^1 \times S^2$ which is homotopy to the identity has to isotope to it?

I guess the question can be asked for all manifolds. But I am particularly interested in $S^1 \times S^2$ right now. Concrete example preferrd.
5
votes
2answers
537 views

Intutive interpretation about Linking forms

Let $M^3$ be a rational homology 3-sphere. (i,e, $M^3$ is closed 3-manifold with $H_{*}(M;Q)=H_{*}(S^3;Q)$ As beautifully explained in Ranicki's Algebraic and Geometry surgery book and Davis-Kirk's ...