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D. Crowley, The classification of highly connected manifolds in dimensions 7 and 15, PhD Thesis, Indiana University, Bloomington, 2002. Available at the arXiv:0203253.

[edit] Corrections

Remark 1.1. The proof of Lemma 2.16 contains a gap in general. There is a surjection \Omega_{2k-1} \colon \Theta_{2k-1}/bP_{2k} \to \Omega_{2k-1}^{HC} but at the time of writing it is in general open whether \Omega_{2k-1} is injective.

However, \Omega_{2k-1} is injective for 2k-1 = 7, 15 by [Wall1962a, Theorem 4] and these are the dimensions in which Lemma 2.16 is used in other parts of the Thesis. For more information about \Omega_{2k-1} see [Stolz1985, Introduction, Theorem B].

Remark 1.2. The expressions for \hat A on p.51 are wrong. A factor of \frac{1}{2^{4j}} needs to be added before each \hat A_j; e.g. A_1 = \frac{-1}{24}p_1.

Remark 1.3. Definition 2.22 of the linking form is not conventional and Remark 2.23 that all three definitions of the linking form agree is incorrect. In fact, the presentation defintion of the linking form differs from the usual definition of the linking form by a sign. Hence, it would fit better with conventions to define

\displaystyle   b(P) : = -b(\lambda) .

For the explanation of this sign see [Alexander&Hamrick&Vick1976, The proof of Theorem 2.1] and [Gordon&Litherland1978, Section 3].

Remark 1.4.Theorem 2.55 is not correct as stated: M must be a closed spin (4k-1)-dimensional rational homology sphere with the extra hypothesis that H^1(M; \Zz/2) = 0. This last hypothesis is needed to ensure that M has a unique spin structure and hence that the manifold X = W \cup W' appearing in the proof admits a spin structure.

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