Suppose we attach an $m$-handle $H^m=D^m\times D^{n-m}$ to an $n$-manifold $M$, with the embedding map $\theta:S^{m-1}\times D^{n-m}\to\partial M$, i.e., $$M\cup_{\theta}H^m=(M\sqcup(D^n\times D^{n-m}))/\sim$$ with the equivalence $(x,y)\sim\theta(x,y)$ for all $(x,y)\in S^{m-1}\times D^{n-m}\subset D^m\times D^{n-m}$.
Then what is the intersection of the $m$-handle $H^m$ and the $n$-manifold $M$? Is it simply a sphere?
I believe the intersection is a sphere; however, I am not too sure how I could show this.
Any help would be much appreciated. Thanks in advance!