||Date of Tuesday
||Manifolds, S, P, T, classification of surfaces. Euler
characteristic. Homogeneous and locally homogeneous metrics.
proof [Francis and Weeks] discusses the classification
theorem for surfaces. Markov's 1958 paper "Insolubility of the
problem of homeomorphy" proves that the homeomorphism problem for
manifolds is undecidable. See also Chapter 9 of the book "Classical
topology and combinatorial group theory" [Stillwell].
||Length elements, geodesics, isometries. Group actions,
reflector points, the four quotients of \(\EE^1\). Tilings,
fundamental domains, classification of isometries of \(\EE^2\), deck
||Orbifolds, discrete groups, and tilings. Classification in
dimension one. Classification of frieze groups. Wallpaper groups,
orbifolds in dimension two.
||In a 2-orbifold we may find: regular points, mirror boundary,
cone points of angle \(2\pi/n\), corner reflectors of angle
\(\pi/n\), regular boundary, and half-mirrored corners. The last
necessarily have angle \(\pi/2\).
A video [Hart]
that mentions frieze patterns and fruit by the foot.
||Interlude on spherical and hyperbolic geometry, the projection
argument. Local groups, orbifold Euler characteristic, orbifold
coverings. Good versus bad.
||Dror Bar-Natan has many lovely pictures
of wallpaper groups. Doug Ravenel's meta-page
is also useful.
||Geometric orbifolds, universal covers, orbifold fundamental
group. Seifert-van Kampen theorem, gluing along 1-orbifolds,
computation of \(\pi_1^\orb\).
||Question from class: Suppose that \(X\) is a complete,
connected, simply-connected, homogeneous metric space. Must \(X\)
be a manifold? Answer: Not necessarily. One example is an \(\RR\)-tree
[Young] arising as an asymptotic cone of the hyperbolic plane. I
don't know a simpler construction.
||Review of 2-orbifolds and orbifolds as quotients of
surfaces. Fibered solid tori and Klein bottles, orientations and
invariants, Seifert fibered spaces, isomorphism, regular and
critical fibers, trivial bundles, a non-trivial example.
||Note that class and office hours on Tuesday are cancelled.
||Review of orbifold fundamental group. Quotient orbifolds of
Seifert fibered spaces. Covers of SFS's and their quotient
orbifolds. Circle actions and orientability of the fibers.
||Calegari makes a connection
between his son's train track set and indiscrete subgroups of
\(\Isom(\EE^2)\). In his set all the segments of track are
exactly one-eighth of a circle. Suppose we had a set with
segments that were, instead, one-quarter of a circle. Show that
the group \(G\) for these pieces is discrete. There are four more
that give discrete groups; find them and their groups.
||Fiber bundles, isomorphism. Sterographic projection, Hopf
fibration, lens spaces. Embeddings and isotopies and their proper
variants. Sections of bundles.
for a visual introduction to sterographic projection and the Hopf
||Disks with handles, \(S^1\)-bundles over surfaces with
boundary, their fundamental groups.
||Friday's class will include a showing of the videos The shape of
space and Not
||Slopes in the torus. Meridional, longitudinal, fiber slopes
for a fibered solid torus. Decomposing Seifert fibered spaces,
sectional slopes, Seifert invariants. Euler number and
||Scott refers to Neumann and Raymond's lecture
notes for many details concerning the Euler number. The notes
by Jankins and Neumann are also useful.
Jeff Weeks has
written several programs for visualising tilings, groups, and
manifolds. We played with his program Curved
spaces in the last class.