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Geometric Hopfian and non-Hopfian situations. (English) Zbl 0607.57015
Geometry and topology, Proc. Conf., Athens/Ga. 1985, Lect. Notes Pure Appl. Math. 105, 157-166 (1987).
[For the entire collection see Zbl 0594.00013.]
Let M be a closed oriented smooth manifold and \(f: M\to M\) a map of degree d. If \(d=\pm 1\) then \(\pi_ 1f: \pi_ 1(M)\to \pi_ 1(M)\) is surjective and the author wishes to know whether:
(1) \(\pi_ 1f\) is injective,
(2) assuming \(\pi_ 1f\) is injective, \(f_*: H_*(M, {\mathbb{Z}}\pi)\to H_*(M, {\mathbb{Z}}\pi)\) is an isomorphism, and
(3) \(f\) is a homotopy equivalence.
The paper is able to show that question 2 (and slight variants thereof) is true, and also shows that question 1, assuming \(d\neq 0\) only, is actually false. The rather cryptic title of the paper refers to so-called Hopfian groups for which every epimorphism is injective.
Reviewer: R.E.Stong

57R19 Algebraic topology on manifolds and differential topology
55M25 Degree, winding number