# On definition of surgery [closed]

I am a beginner in 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?

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.