Geometric Hopfian and Non-Hopfian Situations

Let Mⁿ be a closed oriented (smooth) manifold of dimension n. Let f : M → M be a map of degree d(f). In this paper we consider the following question:

Q

If d(f) = ±1, is f a homotopy equivalence?

This question naturally divides into two sub-questions:

Q1

If d(f) = ±1, is π₁f : π₁(M) → π₁(M) injective?

Observe that π₁f is always surjective (otherwise f would factor through a proper covering of M and hence could not be of degree ±1). Question Q1 was raised by H. Hopf around 1931 (see [10]) and gave rise to the concept of hopfian and non-hopfian groups (a group G is hopfian if any epimorphism of G onto itself is injective).

Q2

If d(f) = ±1 and π₁f is injective (and hence bijective), is f_* :H_* (M; ℤ_π) → H_* (M; ℤ_π) an isomorphism? (π = ℤπ₁(M)).

The answer "yes" to both Questions Q1 and Q2 clearly implies the answer "yes" to Question Q.

Questions Q1 and Q2 may be illuminated by considering natural generalzations of them to maps of non-zero degree. Let d = d(f) and denote by ∧ the ring ℤ [1/d].

Q1a

If d ≠ 0, is π₁f : π₁(M) → π₁(M) injective?

Q2a

If d ≠ 0 and π₁f is injective, is f_* : H_*(M; ∧π) → H_*(M; ∧π) an isomorphism?

158Q2b

If d ≠ 0 and π₁f is bijective, is f_* : H_*(M; ℤπ) → H_*(M; ℤπ) an isomorphism?