Massey Products and k-Equal Manifolds

On Lusternik-Schnirelmann category and topological complexity of no k-equal manifolds

“…Thus, Theorem 1.1 describes cat and TC invariants for manifolds that (as far as it is currently known) might fail to be rationally formal. For instance, while the work in [6] shows that M (3) 2 (n) is rationally formal if and only if n ≤ 6, our calculations show that the equality TC s M (3) 2 (n) = s n k holds for n ≤ 12 with the possible exception of n = 11.…”

“…In fact, non-trivial Massey products holding in these spaces would seem to play a key role in the complete determination of their TC sinvariants. In this direction it is interesting to remark that the hypothesis coming from [6] as a sufficient condition for the rational formality of M (k) 2 (n). For instance, with k = 4, (2) amounts to n ≤ 10, while (1) holds for n ≤ 24, except for n ∈ {19, 22, 23}.…”

“…It is known from [6] that not all manifolds M (k) d (n) are rationally formal when d = 2. In fact, non-trivial Massey products holding in these spaces would seem to play a key role in the complete determination of their TC sinvariants.…”

“…H * (M (3) 2 ( 6)) rank 6)) 20 H 4 (M (3) 2 ( 6)) 45 H 5 (M (3) 2 ( 6)) 36 H 6 1 (M (3) 2 ( 6)) 10 H 6 2 (M (3) 2 ( 6)) 10 H 7 (M (3) 2 ( 6)) 10…”

“…From those numbers, the greatest value is forced to be in a round vertex so it only remains to determine the value of the other round vertex. Therefore there are 6 4 3 1 = 45 basis elements in H 4 (M (3) 2 ( 6)).…”