A consequence of simply connectedness

Question

Is it true that if a manifold is simply connected then its tangent bundle is simply connected?

Answer 1

The tangent bundle is homotopy equivalent to the manifold and therefore has the same fundamental group as the manifold. In particular, they are simply connected simultaneously.

Answer 2

Yes. This follows from a portion of the long exact sequence of a fibration: the tangent bundle $TM$ of a $d$-dimensional manifold $M$ is a fibration with fiber $\mathbb{R}^d$ and so there is an exact sequence $$\pi_1(\mathbb{R}^d) \to \pi_1(TM) \to \pi_1(M) $$