Abstract
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.
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Kreck, M., Lück, W. & Teichner, P. Counterexamples to the Kneser conjecture in dimension four. Commentarii Mathematici Helvetici 70, 423–433 (1995). https://doi.org/10.1007/BF02566016
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DOI: https://doi.org/10.1007/BF02566016