Suppose that we are given a smooth, closed, connected homology sphere $M^n \subset \mathbb{R}^{n+1}$ with $n \geq 6$. I want to kill some elements in the fundamental group of $M$ by surgeries. As far as I understand it correctly, this is done by the following procedure:
- Choosing a representative $S^1 \longrightarrow M$ of the element I want to kill (which can be chosen as an embedding)
- Extending this to an embedding $\varphi : S^1 \times D^{n-1} \longrightarrow M$ and deleting the interior of the image in $M$
- Gluing in a copy of $D^2 \times S^{n-2}$ along the boundary of $M \setminus \mathrm{int}(\varphi (S^1 \times D^{n-1}))$
I think I understand (via a Seifert-van-Kampen argument) why this procedure really kills the chosen homotopy class in the fundamental group.
My question is: How can I guarantee that I can represent my homotopy class in Step 1 by an embedding and why does the extension to an embedding of $S^1 \times D^{n-1}$ in Step 2 really exist? Also, can this extension in Step 2) be done for embeddings of $k$-spheres in arbitrary dimension $k \leq n$?
Any help or literature where I can look things up are greatly appreciated, thank you in advance.
EDIT: As suggested in the comments, I looked at the tubular neighborhood theorem. However, I don't understand how this directly delivers the embeddings. In the literature, I found the condition that the normal bundle of the embedded $S^k$ has to be trivial. Can this be shown for the case at hand?