A series of complexes Q p indexed by all primes p is constructed with cat Q p = 2 and cat Q p ×S n = 2 for either n ≥ 2 or n = 1 and p = 2. This disproves Ganea's conjecture on LS category, or Lusternik-Schnirelmann category.
There is a problem with the proof of Theorem 1.13 of [2] which states that for a fibrewise well-pointed space X over B, we have cat B B (X) = cat * B (X) and that for a locally finite simplicial complex B, we have T C(B) = T C M (B). While we still conjecture that Theorem 1.13 is true, this problem means that, at present, no proof is given to exist. Alternatively, we show the difference between two invariants cat * B (X) and cat B B (X) is at most 1 and the conjecture is true for some cases. We give further corrections mainly in the proof of Theorem 1.12.
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