Symmetries of spheres and group cohomology
Supervisor: Bj酶rn Dundas, email: bjorn.dundas math.uib.no
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Prerequisites: MAT220 is essential. You must also have either taken MAT242 or follow MAT243 in parallel with the project.
Description: Symmetries are a pervasive theme in mathematics. A (banal) example: a regular triangle has a symmetry given by the group of order three consisting of rotations by the angle 2蟺/3.
Another example: in quantum computing, a "pure" state is described by a point on the 3-sphere, but two pure states are in practice equal if they differ by only a "phase shift". This gives a full rotational symmetry, so the state space is only a 2-sphere!
A third example: apparent symmetries in the background radiation in the universe has given rise to the hypothesis that space is shaped like the "Poincar茅 sphere", what you get if you identify points on a 3-sphere in accordance with a certain symmetry of a group of order 120 (the so-called binary icosahedral group).
In this project, we begin with the following question: which finite symmetries are found on spheres? This question was answered by Madsen, Thomas and Wall. The proof contains many themes that are beyond the scope of a bachelor project, so we shall concentrate on the result and the interaction between group theory and geometry.
The goal of the project is that the student shall calculate the group homology for some simple groups acting on spheres.
What you will learn:
- cohomology of groups and methods for calculating these,
- group theory (Sylow subgroups, etc. See Fraleigh's book, which is used in MAT220),
- covers (dealt with in MAT242),
- smooth structures (dealt with in MAT243) and symmetries.
References:
[1] Brown, Kenneth S. . Corrected reprint of the 1982 original. Graduate Texts in Mathematics, 87. , New York, 1994. x+306 pp. ISBN: 0-387-90688-6.
[2] Milnor, John, . Amer. J. Math. 79 (1957), 623鈥�630.
