In this paper, we employ the loop group method to study construction of minimal Lagrangian surfaces in complex projective plane for which surface is contractible. We present several new classes $\mathbb{C} P^2$.

We give an example of a co-compact Kleinian group F which contains a subgroup F 0 having the property that E~/Fo is contractible but not simply connected at infinity.

We consider the configuration space of the Skyrme model and give a simple proof that loops generated by 2π-rotations are contractible in the even-, and noncontractible in the odd-winding-number sectors.

The vortex solution (Z-string) of the electroweak interactions can be interpreted as the 2-dimensional sphaleron at the top of a non-contractible sphere. The same holds for another type of solution, the W -string.

We prove the existence of a positive solution for nonlocal problems involving the fractional Laplacian and a critical growth power nonlinearity when the equation is set in a suitable contractible domain.