I am a beginner in [surgery theory](https://en.wikipedia.org/wiki/Surgery_theory). I have started learning with ALGEBRAIC AND GEOMETRIC SURGERY by Andrew Ranicki.


On page 4 of the book he defines surgery :


**Denition 1.2** A surgery on an $m$-dimensional manifold $M^m$ is the procedure
of constructing a new $m$-dimensional manifold
$$M^{\prime m} =cl.(M\setminus S^n \times D^{m-n})\cup\_{S^{n}\times S^{m-n-1}} D^{n+1}\times S^{m-n-1} $$


by cutting out $S^n \times D^{m-n}\subset M$ and replacing it by $D^{n+1}\times S^{m-n-1}$. The surgery **removes** $S^n \times D^{m-n}\subset M$ and **kills** the homotopy class $S^n \to M$ in $\pi\_n (M)$.


**Question 1**: What is role of $S^{n}\times S^{m-n-1}$ as the subscript of $\cup$?


**Question 2**: I cannot understand the meaning of "it **kills** the homotopy class $S^n \to M$ in $\pi\_n (M)$." Can anyone explain to me?


Thanks in advance.



On definition of surgery
1. The "killing" terminology in surgery is the manifold version of the killing of homotopy classes by attaching cells: for any space $X$ the space $Y=X\cup\_fD^{n+1}$ obtained from $X$ by attaching an $(n+1)$-cell along a map $f:S^n \to X$ has $n$th homotopy group $\pi\_n(Y)=\pi\_n(X)/\langle [f] \rangle$, so the group morphism $\pi\_n(X) \to \pi\_n(Y)$ induced by the inclusion $X \to Y$ sends the homotopy class $[f] \in \pi\_n(X)$ to $0 \in \pi\_n(Y)$, i.e. "kills" it. The killing of homotopy classes by attaching cells is a method for constructing new spaces with particular homotopy theoretic properties (e.g. the Eilenberg-MacLane spaces $K(\pi,n)$) which was developed in the 1940's and 1950's, notably by Cartan and Serre.
2. Milnor's classic 1961 paper "A procedure for killing homotopy groups of differentiable manifolds"
is available from <http://www.maths.ed.ac.uk/~aar/papers/milnorsurg.pdf>
3. I recommend using Google to search for surgery references (such as 2.)


