Fibrewise construction applied to Lusternik-Schnirelmann category

The Lusternik-Schnirelmann category of 𝑆𝑝(3)

“…This invariant was introduced by F. Laudenbach and J.-C. Sikhorav in [12]. Our proof shows that the category of the space Sp(3) coincides with the cone length of Sp(3) and with a stabilized version of the category, denoted Qcat(Sp(3)); see [17,25]. From the main theorem of P.-M. Moyaux and L. Vandembroucq in [15] we know that Crit(Sp(3)) − 1 is less than the cone length and is bounded below by Qcat.…”

Lusternik-Schnirelmann category and products of local spaces

“…Since the proof of 5.7 is central for the whole paper, this shows again the intimate relation between cat and holonomy. In the same direction, in [ST4] certain fibrations which are characterized by means of their holonomy representation were found to detect cat in the same way as the Ganea fibration.…”

“…Let R * , r be a tame ring system such that R ⊇ R i for i k and write r + k = m. The following theorem is implicit in [ST1], because it was proved in Theorem 4 that the fibration corresponding to j (see below) is an n-LS-fibration in the sense of [ST4]. By definition, such fibrationsp n come with maps s, t over the base from and to the n-th Ganea fibration p n such that p n • t p n andp n • s p n .…”

“…The theorem is a direct consequence of 6.2 and 6.3. We have to rely on certain results of Scheerer and Tanré in [ST1,ST4]. The proof would be straightforward, if ΛX → ΛX ⊗ ΛY m−1 represented the (m − 1)-Postnikov section of the n-th Ganea fibration over Z.…”

Implications of the Ganea condition

“…In fact, it was only recently that techniques were developed which were powerful enough to identify a space which does not satisfy the Ganea condition [8] (see also [9,12]). It is still not well understood exactly which spaces X do not satisfy the Ganea condition, although it has been conjectured that they are precisely those spaces for which cat(X) is not equal to the related invariant Qcat(X) (see [14,17]). …”