# Template:Crowley2002

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 but at the time of writing it is in general open whether is injective.

However, is injective for 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 see [Stolz1985, Introduction, Theorem B].

**Remark 1.2.** The expressions for on p.51 are wrong. A factor of needs to be added before each ; e.g. .

**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

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: must be a closed spin -dimensional rational homology sphere *with the extra hypothesis* that . This last hypothesis is needed to ensure that has a unique spin structure and hence that the manifold appearing in the proof admits a spin structure.