Commentarii Mathematici Helvetici

, Volume 70, Issue 1, pp 423–433 | Cite as

Counterexamples to the Kneser conjecture in dimension four

  • Matthias Kreck
  • Wolfgang Lück
  • Peter Teichner


We construct a connected closed orientable smooth four-manifold whose fundamental group is the free product of two non-trivial groups such that it is not homotopy equivalent toM 0#M 1 unlessM 0 orM 1 is homeomorphic toS 4. LetN be the nucleus of the minimal elliptic Enrique surfaceV 1(2, 2) and putM=N∪ ∂NN. The fundamental group ofM splits as ℤ/2 * ℤ/2. We prove thatM#k(S 2×S2) is diffeomorphic toM 0#M 1 for non-simply connected closed smooth four-manifoldsM 0 andM 1 if and only ifk≥8. On the other hand we show thatM is homeomorphic toM 0#M 1 for closed topological four-manifoldsM 0 andM 1 withπ 1(Mi)=ℤ/2.


Fundamental Group Free Product Homotopy Equivalent Regular Neighborhood Simple Homotopy 
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Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

  • Matthias Kreck
    • 1
    • 2
    • 3
    • 4
  • Wolfgang Lück
    • 1
    • 2
    • 3
    • 4
  • Peter Teichner
    • 1
    • 2
    • 3
    • 4
  1. 1.Fachbereich MathematikJohannes Gutenberg-UniversitätMainzBundesrepublik Deutschland
  2. 2.Mathematisches Forschungsinstitut OberwolfachOberwolfach-WalkeBundesrepublik Deutschland
  3. 3.University of CaliforniaSan Diego
  4. 4.Department of MathematicsLaJollaU.S.A.

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