| The last version of the lecture notes, as of 19th February 2018, can be found here. These are lecture notes about geometry and topology leading to the initial study of surface group representations and the de nition of Higgs bundle. They reflect the contents of this course and are better understood with the images from the lectures. |
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For an alternative presentation of manifolds in the same spirit, look at Sections 1 and 2 of [Hit12]. For more on the classification of surfaces, look at Chapter I of [Mas91]. For more on the fundamental group and Seifert-van Kampen theorem, look at Chapters II-IV of [Mas91] or Sections 1.1 and 1.2 of [Hat17]. More references can be found in the lecture notes. The advanced references will be useful once we have worked on the prerequisites.
More background material
[Hit12] N. Hitchin. Differentiable manifolds (lecture notes). Available online.
[Mas91] W.S. Massey. A Basic Course in Algebraic Topology. Graduate Texts in Mathematics. Springer-Verlag, 1991. Available in the library (514.2 MAS) or borrow a copy from me.
[Hat17] A. Hatcher. Algebraic Topology. Available online.
Advanced references
[Bra12] S. B. Bradlow. Introduction to Higgs bundles. Slides for the GEAR Retreat 2012 mini-course. University of Illinois at Urbana-Champaign. Available online.
[Got08] P. Gothen. Higgs bundles. Slides for the International School on Geometry and Physics: moduli spaces in geometry, topology and physics, CIEM, 2008. Available online.
[Got11] P. Gothen. Representations of surface groups and Higgs bundles. Lecture notes for the School on Moduli Spaces at the Newton Institute, 2011. Available online.
[BGGW] S.B. Bradlow, O. Garcia-Prada, W. Goldman, A. Wienhard. Representations of surface groups: background material for AIM workshop. Available online.
[BGG] S. Bradlow., O. Garcia-Prada, P. B. Gothen. WHAT IS... a Higgs Bundle? Notices Amer. Math. Soc. 54 (2007), no. 8, 980-981. Available online.
Further reading:
[Wen12] R. Wentworth. Higgs Bundles and Local Systems on Riemann Surfaces. Lecture notes for the 3rd International School on Geometry and Physics at the CRM, 2012. Geometry and Quantization of Moduli Spaces, 165{219, Springer 2016. Available online.
The original papers:
[Hit87] N. J. Hitchin. The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3), 55(1):59-126, 1987. Available online.
[Sim92] C. T. Simpson. Higgs bundles and local systems. Inst. Hautes Etudes Sci. Publ. Math., (75):5-95, 1992. Available online.