Let be the mapping cone of the map defined by .

How do you compute the homology groups of ? What about the homology groups of the Universal covering of ?

I know that the mapping cone of , is defined to be the quotient of the mapping cylinder of with .

Or we can say,

Given a map , the mapping cone is defined to be the quotient of the topological space of with respect to the equivalence relation , on . Here denotes the unit interval with its standard topology.

But I am not sure how to start using this definition of the mapping cone, to find the homology groups of and of its universal cover.