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在Berger 對非對稱流形Holonomy group 的分類中，Spin(7) 為一個特例，八維流形可能會有Spin(7) 的Holonomy group，而第一個緊緻且有Spin(7) Holonomy group 的流形是由Dominic Joyce 所構造，在這篇文章中會討論Joyce 構造流形的過程。

In Berger’s classification of holonomy groups of non-symmetric manifolds, Spin(7) is a special case. The first compact manifold with holonomy Spin(7) is constructed by Dominic Joyce in 1996. In this article, we will discuss Joyce’s construction of such manifolds.

口試委員審定書i
誌謝ii
中文摘要iii
Abstract iv
1 Introduction 1
1.1 Holonomy groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Spin(7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Forms and Spin(7)-structures . . . . . . . . . . . . . . . . . . . . 4
1.4 Eguchi-Hanson space and Kummer construction . . . . . . . . . . 5
1.5 Sobolev space, Sobolev norm . . . . . . . . . . . . . . . . . . . . 7
2 Crucial Theorems and Steps to construct manifolds with holonomy
Spin(7) 7
2.1 Crucial Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Steps to construct manifolds with holonomy Spin(7) . . . . . . . 9
3 Construction of 8-manifolds 9
3.1 Resolving singularities of orbifolds . . . . . . . . . . . . . . . . . 9
3.2 8-manifolds with holonomy Spin(7) . . . . . . . . . . . . . . . . . 12
References 16

References
[1] Dominic D.Joyce Compact 8-manifolds with holonomy Spin(7). (1996)
[2] Dominic D.Joyce Compact Manifolds with Special Holonomy. (2000)
[3] Dominic D.Joyce Riemannian Holonomy Groups and Calibrated Geometry.
(2007)
[4] Dominic D.Joyce Compact riemannian 7-manifolds with holonomy G2 I.
(1996)
[5] Dominic D.Joyce Compact riemannian 7-manifolds with holonomy G2 II.
(1996)
[6] Dominic D.Joyce Compact Riemannian Manifolds with Exceptional Holonomy.
(1999)
[7] Christine Taylor Compact Manifolds with Holonomy Spin(7). (1996)
[8] Simon Salamon Riemannian geometry and holonomy groups. (1989)
[9] Robert L.Bryant Metric with exceptional holonomy. (1987)
[10] Anthony W. Knapp Lie Groups Beyond an Introduction, Second Edition.
(2002)
[11] Raoul Bott and Loring W. Tu Differential Forms in Algebraic Topology.
(1982)
[12] Claude Chevalley and Samuel Eilenberg Cohomology Theory of Lie Groups
and Lie Algebras. (1948)
[13] John W. Milnor and James D. Stasheff Characteristic classes. (1974)
[14] John M. Lee Introduction to smooth manifolds. (2002)
[15] Jurgen Jost Riemannian Geometry and Geometric Analysis. (2008)
[16] Phillip Griffiths and Joseph Harris Principles of Algebraic Geometry. (1994)