Template:Lück2001

W. Lück, A basic introduction to surgery theory, 9 (2001), 1–224. Available from the author's homepage. MR1937016 (2004a:57041) Zbl 1045.57020

Comments and corrections for this work follow organised by chapter.

3 Chapter 3

page 57:

• In equation (3.22) one should mention that $\xi\times\eta$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\xi\times\eta$ is a bundle over the product $X_1\times X_2$$X_1\times X_2$.
• Typos at the bottom: write $\mathrm{Th}(\nu(M))$$\mathrm{Th}(\nu(M))$ instead of $\mathrm{Th}(TM)$$\mathrm{Th}(TM)$ three times.
• Typo in line 11: write $\mathrm{Th}(\xi)$$\mathrm{Th}(\xi)$ instead of $\mathrm{Th}(E)$$\mathrm{Th}(E)$.

page 58:

• typo in line 8 from the bottom: write $f$$f$: $M\to X$$M\to X$ instead of $g$$g$: $M\to X$$M\to X$.
• In the proof of Theorem 3.26: $X$$X$ is a CW complex, not a manifold. Why is $f^{-1}(X)$$f^{-1}(X)$ a manifold? What is the exact definition of transversality here?

page 59:

• The word 'homomorphism' in line 3 is confusing, since the group structure on $\Omega_n(\gamma_k)$$\Omega_n(\gamma_k)$ is not clear.
• Typo in line 9: write $V_{k+1}\circ\Omega_n(\overline{i_k})$$V_{k+1}\circ\Omega_n(\overline{i_k})$ instead of $V_{k+1}\circ i_k$$V_{k+1}\circ i_k$.
• Typo in the definition of $s_k$$s_k$: write $\pi_{n+k+1}(\mathrm{Th}(\overline{i_k}))$$\pi_{n+k+1}(\mathrm{Th}(\overline{i_k}))$ instead of $\pi_{n+k}(\mathrm{Th}(\overline{i_k}))$$\pi_{n+k}(\mathrm{Th}(\overline{i_k}))$.

page 60:

• Typo in line 3: write $\pi_{n+k+1}(\sigma_k)$$\pi_{n+k+1}(\sigma_k)$ instead of $\pi_{n+k}(\sigma_k)$$\pi_{n+k}(\sigma_k)$.

page 61:

• Typo in line 1: write $p_0$$p_0$: $E_0\to X_0$$E_0\to X_0$ instead of $p_0$$p_0$: $E_0\to X$$E_0\to X$.
• Typo in line 1: write $p_1$$p_1$: $E_1\to X_1$$E_1\to X_1$ instead of $p_1$$p_1$: $E_1\to X$$E_1\to X$.

page 62:

• Typo in the assertion of Theorem 3.38: write $\pi_{n+k}(\mathrm{Th}(p_0))\to\pi_{n+k}(\mathrm{Th}(p_1))$$\pi_{n+k}(\mathrm{Th}(p_0))\to\pi_{n+k}(\mathrm{Th}(p_1))$ instead of $\pi_{n+k}(\mathrm{Th}(p_0))\to\pi_{n+k}(\mathrm{Th}(p_0))$$\pi_{n+k}(\mathrm{Th}(p_0))\to\pi_{n+k}(\mathrm{Th}(p_0))$.

page 64:

• Typo in Lemma 3.40: write $c_1$$c_1$: $S^{n+k}\to\mathrm{Th}(p_1)$$S^{n+k}\to\mathrm{Th}(p_1)$ instead of $c_1$$c_1$: $S^{n+k}\to\mathrm{Th}(p_0)$$S^{n+k}\to\mathrm{Th}(p_0)$.
• Lemma 3.40 uses the notion of degree for maps between arbitrary Poincare complexes. It would be helpful to provide a general definition of degree in the lecture notes before this lemma.

page 65:

• Typo in line 3: write $\mathrm{Th}(\xi_0)$$\mathrm{Th}(\xi_0)$ instead of $\mathrm{Th}(E_0)$$\mathrm{Th}(E_0)$.
• In the three lines after 'One can arrange that these stabilization maps...' it would be useful to have a more precise explanation, why one should use cofibrations.
• Typo in line 13 from the bottom: write Spivak normal $(k-1)$$(k-1)$-fibration instead of Spivak normal $(k-1)$$(k-1)$-bundle.
• Remark 3.44: $G/O$$G/O$ is used here but the definition of $G/O$$G/O$ is after the Remark. It is better to give the definition of $G/O$$G/O$ already here. Also it does not become clear what $B(G/O)$$B(G/O)$ is.

page 67:

• In Definition 3.46: If two vector bundles $\xi$$\xi$ and $\xi'$$\xi'$ are isomorphic but not equal, the corresponding normal $k/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_m8nl9a$$k$-maps should nevertheless be identified. However in this definition only equal vector bundles are identified. See Part 1 on the following discussion page for some suggestions on this definition.
• After Definition 3.46 the notion of degree of a map between Poincare pairs is used. It would be helpful to provide a general definition in the lecture notes before.
• One would like to have more details of the proof of Theorem 3.48. This theorem is an important step and thus should be treated more precisely.

page 68:

• In Definition 3.50 one should consider closed manifolds, not closed oriented manifolds. Also there are some typos in the definition of bordism after Definition 3.50. See Part 2 on the following discussion page for some suggestions on this definition.

page 69:

• In line 3 one should consider closed manifolds, not closed oriented manifolds.

4 Chapter 4

Example 4.20: the surgery is performed on the zero element in $\pi_k(f)$$\pi_k(f)$ not in $\pi_{k+1}(f)$$\pi_{k+1}(f)$.