The "cone" part is contractible to the "point" , so its homology is easy to compute. I'd guess that the Mayer-Vietoris sequence might be the way to proceed, but my algebraic topology class just ended and we didn't do mapping cones.
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.