Mapping Class group of 3-manifold $S^1\times S^1\times D^1$

Question

Is Mapping Class group of $S^1\times S^1\times D^1$ trivial? $D^1$ stands for 1-dimensional disc. Thanks

Answer 1

Far from being trivial, it is an infinite group.

To see this, first you need the theorem from surface theory that the canonical homomorphism $$\text{MCG}(S^1 \times S^1) \to \text{Aut}(H_1(S^1 \times S^1)) \approx \text{Aut}(\mathbb{Z}^2) \approx GL(2,\mathbb{Z}) $$ is an isomorphism.

Next, the projection map $p : (S^1 \times S^1) \times D^1 \to S^1 \times S^1$ is a homotopy equivalence, so it induces a homomorphism $$\text{MCG}((S^1 \times S^1) \times D^1) \to \text{Aut}(H_1(S^1 \times S^1)) \approx GL(2,\mathbb{Z}) $$

Finally, this homomorphism is onto because every homeomorphism $f : S^1 \times S^1 \to S^1 \times S^1$ extends to a homeomorphism $$f \times \text{Id} : (S^1 \times S^1) \times D^1 \to (S^1 \times S^1) \times D^1 $$