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?

Thanks in advance.

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    $\begingroup$ Q1: You are taking the union of two manifolds, and their common boundary is this product of spheres. This notation is indicating that you're gluing the manifolds together along this product of spheres. Q2: the boundary includes into the manifold, so there's a map of homotopy groups, and he's saying after you've formed this manifold, a particular sphere becomes trivial, since it bounds a disc. $\endgroup$ Jul 14 '13 at 8:46
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    $\begingroup$ @RyanBudney Why not post your comment as an answer? $\endgroup$
    – C.S.
    Jul 14 '13 at 8:46
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    $\begingroup$ You might prefer to start with Wolfgang Luck's notes: $\endgroup$ Jul 14 '13 at 10:13
  • $\begingroup$ Hi @Sepideh: These are good questions, but not really "research level". I recommend asking them on math.stackexchange, where there are many experts and the scope of the site is defined more broadly. $\endgroup$ Jul 14 '13 at 17:36
  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.)

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