Let $G=SU(2)$. Let $\mu:G^4\to G$ be the product of commutators map $\mu(x,y,x',y')=[x,y][x',y']$. Let $M$ be the $9$-manifold $\mu^{-1}(-1)$ on which $G$ acts by componentwise conjugation. The $6$-manifold $A:=\mu^{-1}/G$ was studied by Atiyah-Bott (1982) who used Morse Theory to compute its cohomology groups as part of a more general program. The manifold $A$ represents the moduli space of flat connections on a punctured $2$-hole torus with holonomy $-1$ on the boundary. It is an example of the symplectic reduction of a quasi-Hamiltonian $G$-space (Alekseev-Malkin-Meinrenken; 1998). The space has also been studied by means of algebraic geometry: an algebraic variety corresponding to $A$ was studied by Newstead (1966) and the ring structure of $H^*(A)$ was computed using algebraic geometry by Thaddeus (1992).

We study $M$, $A$, and preimages of the commutator map $Comm:G^2\to G$ using elementary techniques and obtain their cohomology using Mayer-Vietoris sequences. For regular values $c$ of $Comm$ we give an explicit homeomorphism $Comm^{-1}(c)\cong PSU(2)=SO(3)=RP^3$. The identity is not a regular value of $Comm$: the space $Comm^1$ of commuting pairs has been studied by Adem-Cohen (2006) who computed its cohomology groups and by Baird-Jeffrey-Selick (2009) who gave the homotopy type of its suspension. In his talk we strengthen these results to include its homotopy type and the ring structure of its cohomology groups. After reproducing the results of Atiyah-Bott and Thaddeus on $H^*(A)$ by elementary methods we use the methods to show that the principal bundle $PSU(2)\to M\to A$ has a local trivialization over two charts and we find the transition function. From this we obtain the cohomology $H^*(M)$.

The talk will emphasize explicit homeomorphisms and computations by elementary methods rather than high-powered machines.