# Interesting results/topics in algebraic topology and representation theory

What are some interesting topics to read up on and research within algebraic topology and/or representation theory, or more ideally, and intersection of both. I have previously spent time looking at representations of the fundamental group (for coursework some time ago), I found it very interesting that we can obtain representations of $\pi_1(X)$ by studying covering spaces of the topological space. However, I could find very little literature about this (if anyone knows anything good please point it out!). I think this relates to monodromy.

While this is an example, I think topics of this flavour are very interesting.

I think my knowledge to begin reading such things is probably close to sufficient. I am quite well versed in differential geometry, topology, cohomology on topological spaces/manifolds, commutative algebra, homotopy theory, representation theory, category theory. I have recently spent much time reading on sheaves too.

Do tell! What do you find interesting? References and texts are welcomed.

• There's an unsolved problem about whether the Burau representation of the braid groups if faithfull for $B_4$. arxiv.org/pdf/1705.02641.pdf
– Javi
May 9, 2018 at 11:38
• Representation theory plays a prominent role in defining and obtaining many gauge-theoretic invariants of 3-manifolds. On the other hand, algebraic topological methods seem to be playing an increasingly important role in this field, and Floer theory, in particular, seems to be profiting from this. Manolescu's paper, arxiv.org/pdf/1708.00289.pdf, makes some specific connections with topics you mention, and you might like to check out more of his work relating to his SWF stable homotopy type. May 10, 2018 at 8:44
• I liked Burt Totaro's book "Group Cohomology and Algebraic Cycles", which I feel has a quite geometric flavor to it.
– user98602
May 10, 2018 at 18:28

Quantum groups and their representations have applications to knot theory, for example it is possible to build the Jone polynomial using $U_q(\mathfrak {sl}_2)$ representations.
Another example is given by the monodromy representation you mentioned, applied to a specific map (I won't discuss the details, see "Four lectures on simple groups and simple singularities" by Slodowy). This gives representation of the Weyl group $W$ in the homology of Springer fibers.
Finally, if you consider sheaf theory as part of topology, there is a theorem of Beilinson-Bernstein relating $D$-modules on the flag variety $G/B$ and representation theory of $\mathfrak g$. Using the abstract Riemann-Hilbert correspondence, you can replace $D$-modules by complexes of sheaves called perverse sheaves. This area is called geometric representation theory.