Abstract
If we want to somehow classify surfaces, we have to deal with the issue of deciding when we consider two surfaces to be equivalent. It seems reasonable to treat homeomorphic surfaces as equivalent, but this leads to the problem of deciding when two surfaces are not homeomorphic, which is a very difficult problem. One way to approach this problem is to forget some of the topological structure of a surface and look for more algebraic objects that can be associated with a surface.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
For fixed u, the map x↦(1 − u)x is a central magnification of center a and ratio 1 − u. This is an affine map, and it can be expressed linearly because the origin has been chosen as the center of this magnification. Thus, (1 − u)γ(t) makes sense.
References
L.V. Ahlfors, L. Sario, Riemann Surfaces, Princeton Math. Series, No. 2 (Princeton University Press, Princeton, 1960)
G.E. Bredon, Topology and Geometry, GTM No. 139. 1st edn. (Springer, New York, 1993)
A. Dold, Lectures on Algebraic Topology, 2nd edn. (Springer, Berlin, 1980)
W. Fulton, Algebraic Topology, A first course, GTM No. 153, 1st edn. (Springer, New York, 1995)
A. Hatcher, Algebraic Topology, 1st edn. (Cambridge University Press, Cambridge, 2002)
W.S. Massey, Algebraic Topology: An Introduction, GTM No. 56, 2nd edn. (Springer, New York, 1987)
W.S. Massey, A Basic Course in Algebraic Topology, GTM No. 127, 1st edn. (Springer, New York, 1991)
J.R. Munkres, Topology, 2nd edn. (Prentice Hall, NJ, 2000)
J.J. Rotman, Introduction to Algebraic Topology, GTM No. 119, 1st edn. (Springer, New York, 1988)
H. Sato, Algebraic Topology: An Intuitive Approach, MMONO No. 183, 1st edn. (AMS, Providence, 1999)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Gallier, J., Xu, D. (2013). The Fundamental Group, Orientability. In: A Guide to the Classification Theorem for Compact Surfaces. Geometry and Computing, vol 9. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34364-3_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-34364-3_4
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34363-6
Online ISBN: 978-3-642-34364-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)