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# Geometric Hopfian and Non-Hopfian Situations

DOI link for Geometric Hopfian and Non-Hopfian Situations

Geometric Hopfian and Non-Hopfian Situations book

# Geometric Hopfian and Non-Hopfian Situations

DOI link for Geometric Hopfian and Non-Hopfian Situations

Geometric Hopfian and Non-Hopfian Situations book

## ABSTRACT

Let M^{n} 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:

Q1If d(f) = ±1, is π_{1}f : π_{1}(M) → π_{1}(M) injective?

Observe that π_{1}f 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).

If d(f) = ±1 and π_{1}f is injective (and hence bijective), is f_{*} :H_{*} (M; ℤ_{π}) → H_{*} (M; ℤ_{π}) an isomorphism? (π = ℤπ_{1}(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].

Q1aIf d ≠ 0, is π_{1}f : π_{1}(M) → π_{1}(M) injective?

If d ≠ 0 and π_{1}f is injective, is f_{*} : H_{*}(M; ∧π) → H_{*}(M; ∧π) an isomorphism?

If d ≠ 0 and π_{1}f is bijective, is f_{*} : H_{*}(M; ℤπ) → H_{*}(M; ℤπ) an isomorphism?