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SUMMARY:Functorial semi-norms and a problem of Hopf for circle bundles
DTSTART;VALUE=DATE-TIME:20180910T090000Z
DTEND;VALUE=DATE-TIME:20180910T100000Z
DTSTAMP;VALUE=DATE-TIME:20210413T161118Z
UID:indico-event-3863@indico.math.cnrs.fr
DESCRIPTION: \n\n\n\n\n\nA long-standing question of Hopf asks whether ev
ery self-map of absolute degree one of a closed oriented manifold is a hom
otopy equivalence. This question gave rise to several other problems\, mos
t notably whether the fundamental groups of aspherical manifolds are Hopfi
an\, i.e. any surjective endomorphism is an isomorphism. Recall that the B
orel conjecture states that any homotopy equivalence between two closed as
pherical manifolds is homotopic to a homeomorphism. In this talk\, we ver
ify a strong version of Hopf's problem for certain aspherical manifolds. N
amely\, we show that every self-map of non-zero degree of a circle bundle
over a closed oriented aspherical manifold with hyperbolic fundamental gro
up (e.g. negatively curved manifold) is either homotopic to a homeomorphi
sm or homotopic to a non-trivial covering and the bundle is trivial. Our m
ain result is that a non-trivial circle bundle over a closed oriented asph
erical manifold with hyperbolic fundamental group does not admit self-map
s of absolute degree greater than one. This extends in all dimensions the
case of circle bundles over closed hyperbolic surfaces (which was shown by
Brooks and Goldman) and provides the first examples (beyond dimension thr
ee) of non-vanishing functorial semi-norms on the fundamental classes of
circle bundles over aspherical manifolds with hyperbolic fundamental group
s.\n\n\n\n\n\nhttps://indico.math.cnrs.fr/event/3863/
LOCATION:IHES Amphithéâtre Léon Motchane
URL:https://indico.math.cnrs.fr/event/3863/
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