Spaces of Locally Convex Curves in S^n and Combinatorics of the Group B_(n+1)^+

Results on the homotopy type of the spaces of locally convex curves on 𝕊 3

“…For n = 3, the result in [18] says that given (z l , z r ) ∈ S 3 × S 3 , the space LS 3 (z l , z r ) is homeomorphic to at least one of the five spaces:…”

“…There we recall some basic notions on the spin group and on signed permutation matrices which will be necessary to explain the Bruhat decomposition of the special orthogonal group and the lifted decomposition to the spin group. This decomposition was already an important tool in [18], and it will also be very important for us.…”

“…We will consider that our curves are smooth, but in the construction, we will not be bothered by the loss of smoothness due to juxtaposition of curves. The class of differentiability is not important: see [18], [17] or [1] for a discussion of this technical point. We need to clarify the notion of multiplicity in this definition.…”

“…the space LS 3 (Q 0 ) has two connected components LS 3 (i, −i) and LS 3 (−i, i). It follows from [18] that for, any Q ∈ SO 4 , the space LS 3 (Q) is homeomorphic to one of these: LS 3 (I), LS 3 (−I) or LS 3 (Q 0 ). For any Q ∈ SO 4 , there is a natural inclusion F (to be described below) of each connected component into Ω(S 3 × S 3 ).…”

“…For any Q ∈ SO 4 , there is a natural inclusion F (to be described below) of each connected component into Ω(S 3 × S 3 ). Furthermore, the inclusions LS 3 (±i, ∓i) ⊂ Ω(S 3 × S 3 ) are homotopy equivalences ( [18]). We prove that the same does not hold for the spaces LS 3 (−1, 1) and LS 3 (1, −1) n :…”