Geometric Group Thoery, Low-dimensional Topology, Contact and Simplectic Geometry

Recommended Texts or Study Materials

Algebraic Topology - Hatcher PDF
Differential Geometry (Manifolds, Vector fields, Differential forms)
Kirby Calculus (Handle decompositions, Heegaard diagrams) Recommended Refernece: 4-Manifolds and Kirby Calculus by Gompf and Stipsicz

Groups, Graphs, and Trees, Meier.
A Course on Geometric Group Theory, Bowditch. (This is available online, they are lecture notes)
Topological Methods in Group Theory, Scott and Wall.
Geometric Group Theory, Papazoglou.

Computing the fundamental group, homology, and cohomology of various spaces. Covering spaces, deck transformations, cyclic covers, etc. Meyer-Vietoris.
Understand and be able to apply Universal Coefficient theorem, Kunneth Formula and Duality theorems

Cayley graphs, quasi-isometry invariants (e.g. Gromov Hyperbolic, Dehn functions, boundary, etc.), CAT(k) spaces, Milnor Svarc Lemma.

Advisor 1	Committee Member 1	Committee Member 2
Dr. Pallavi Dani	Dr. Shea Vela-Vick	Dr. Dan Cohen

Advisor 1	Committee Member 1	Committee Member 2
Dr. Oliver Dasbach	Dr. Shea Vela-Vick	Dr. Pallavi Dani
Dr. Shea Vela-Vick	Dr. Scott Baldridge	Dr. Pallavi Dani

Prove that \(\mathbb{R}^m\cong \mathbb{R}^n \) iff \(m=n\). Next, compute the homology of \(S^n\).
Compute the homology of \(S^2\wedge S^1\).
Pick a graph of your choice (or for some CW complex), draw the 2-fold covering space, draw the universal covering space, What are the deck transformations?
Show why \( \mathbb{C}P^{2} \) is not the same as its orientation reverse using cohomology.
Show that for any connected closed orientable \(n\) manifold \(M\) there is a degree 1 map \(M \rightarrow S^n\)
\( \mathbb{R}P^3 \) is not homotopy equivalent to \( \mathbb{R}P^2 \vee S^3 \)
Show that a compact manifold does not retract onto its boundary
Show that for a degree 1 map \( f : M \ rightarrow N \) of connected closed orientable manifolds, the induced map on fundamental group is surjective, hence also the one induced in homology
Are there degree 1 maps from closed oriented genus \(g\) surface \(\Sigma_g \to \Sigma_h\) for arbitrary \(g\) and \(h\)? restrictions on \(g\) and \(h\)?
What does a regular covering space look like? Specifically, I had a base space of \(S^1\wedge S^1\), which has fundamental group \(F_2=\langle a,b \rangle\) and I constructed the covering spaces corresponding to the subgroups \(\langle a \rangle\) and \(\langle a^2, ab, ab^{-1} \rangle\) and had to explain why the latter covering space "looked" normal while the former covering space did not.

Show that the genus 3 surface is a covering space of the genus 2 surface. (This was motived by showing that all fundamental groups of orientable surfaces with genus \( \geq 2 \) are Gromov hyperbolic).
Show that BS(1,2) is not a hyperbolic group (i.e. construct its Cayley graph and show its Dehn function has hyperbolic growth.)