The set of 3-manifolds with the same homology as the 3-dimensional sphere, modulo an equivalence relation called *homology cobordance*, forms a group.The additive structure of this group is given by taking connected sum. This group is called the *homology cobordism group *and plays a special role in low dimensional topology and knot theory. In this talk, I will explain how one can construct a family of invariants of the homology cobordism group by applying ideas from min-max theory in Floer theory. The relationship between these invariants and the Froyshov’s invariant will be discussed. I will also talk about some topological applications.