Template:Lück2001

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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.

Contents

[edit] 1 Chapter 1

[edit] 2 Chapter 2

[edit] 3 Chapter 3

page 57:

  • In equation (3.22) one should mention that \xi\times\eta is a bundle over the product X_1\times X_2.
  • Typos at the bottom: write \mathrm{Th}(\nu(M)) instead of \mathrm{Th}(TM) three times.
  • Typo in line 11: write \mathrm{Th}(\xi) instead of \mathrm{Th}(E).

page 58:

  • typo in line 8 from the bottom: write f: M\to X instead of g: M\to X.
  • In the proof of Theorem 3.26: X is a CW complex, not a manifold. Why is 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) is not clear.
  • Typo in line 9: write V_{k+1}\circ\Omega_n(\overline{i_k}) instead of V_{k+1}\circ i_k.
  • Typo in the definition of s_k: write \pi_{n+k+1}(\mathrm{Th}(\overline{i_k})) instead of \pi_{n+k}(\mathrm{Th}(\overline{i_k})).

page 60:

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

page 61:

  • Typo in line 1: write p_0: E_0\to X_0 instead of p_0: E_0\to X.
  • Typo in line 1: write p_1: E_1\to X_1 instead of p_1: 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)) instead of \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: S^{n+k}\to\mathrm{Th}(p_1) instead of c_1: 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) instead of \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)-fibration instead of Spivak normal (k-1)-bundle.
  • Remark 3.44: G/O is used here but the definition of G/O is after the Remark. It is better to give the definition of G/O already here. Also it does not become clear what B(G/O) is.

page 67:

  • In Definition 3.46: If two vector bundles \xi and \xi' are isomorphic but not equal, the corresponding normal 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.

[edit] 4 Chapter 4

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

[edit] 5 Chapter 5

[edit] 6 Chapter 6

[edit] 7 Chapter 7

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