Mathematics — Graph Theory · homotopy-equivalence-universal-cover-homology
Problem
Show that a map f:X→Y of connected CW complexes is a homotopy equivalence if it induces an isomorphism on π1 and if a lift f~:X~→Y~ to the universal covers induces an isomorphism on homology. [The latter condition can be restated in terms of homology with local coefficients as saying that f∗:H∗(X;Z[π1X])→H∗(Y;Z[π1Y]) is an isomorphism; see §3.H.]