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Andrew Ranicki
Andrew Ranicki
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  1. The repetition of "and" in the quotation is a transcription error, and not in the original!
  2. 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.
  3. 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
  4. I recommend using Google to search for surgery references (such as 32.)
  1. The repetition of "and" in the quotation is a transcription error, and not in the original!
  2. 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.
  3. 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
  4. I recommend using Google to search for surgery references (such as 3.)
  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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Andrew Ranicki
Andrew Ranicki
  • 3.9k
  • 1
  • 36
  • 26
  1. The repetition of "and" in the quotation is a transcription error, and not in the original!
  2. 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.
  3. Milnor's classic 1961 paper "A procedure for killing homotopy groups of differenntiabledifferentiable manifolds" is available from http://www.maths.ed.ac.uk/~aar/papers/milnorsurg.pdf
  4. I recommend using Google to search for surgery references (such as 3.)
  1. The repetition of "and" in the quotation is a transcription error, and not in the original!
  2. 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.
  3. Milnor's classic 1961 paper "A procedure for killing homotopy groups of differenntiable manifolds" is available from http://www.maths.ed.ac.uk/~aar/papers/milnorsurg.pdf
  4. I recommend using Google to search for surgery references (such as 3.)
  1. The repetition of "and" in the quotation is a transcription error, and not in the original!
  2. 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.
  3. 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
  4. I recommend using Google to search for surgery references (such as 3.)
Source Link
Andrew Ranicki
Andrew Ranicki
  • 3.9k
  • 1
  • 36
  • 26

  1. The repetition of "and" in the quotation is a transcription error, and not in the original!
  2. 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.
  3. Milnor's classic 1961 paper "A procedure for killing homotopy groups of differenntiable manifolds" is available from http://www.maths.ed.ac.uk/~aar/papers/milnorsurg.pdf
  4. I recommend using Google to search for surgery references (such as 3.)