Rational homotopy models for two-point configuration spaces of lens spaces

Abstract

We study the algebraic topology of configuration spaces as interesting objects in their own right and with the goal of constructing invariants for topological manifolds. We calculate the complete Massey product structure for the universal cover of the space of two point configurations in a three-dimensional lens space. We then construct rational homotopy models for these spaces and calculate the rational homotopy groups.

On the deleted squares of lens spaces

“…This result prompted a natural question pertaining specifically to lens spaces: does the homotopy type of configuration spaces distinguish all lens spaces up to homeomorphism ? This question was studied by Miller [9] for two-point configuration spaces, also known under the name of deleted squares. Miller extended the Massey product calculation of [8] to arbitrary 2000 Mathematics Subject Classification.…”

On the deleted squares of lens spaces

“…The homotopy equivalence f : X 0 → X 0 of this theorem lifts to a homotopy equivalencef :X 0 →X 0 of the universal covering spaces of the type studied by Longoni and Salvatore [8] and Miller [9]. The homotopy equivalencef naturally possesses equivariance properties made explicit by the knowledge of the induced map f * on the fundamental groups.…”

“…The Massey products (9) were calculated by Miller [9,Theorem 3.33] for all L(p, q) such that 0 < q < p/2 using the intersection theory of [8]. The theorem below summarizes Miller's calculation.…”

“…According to the above theorem, 1, 1, t 2 = t + t 2 , 1, 1, t 3 = t 3 + t 4 , t, 1, t 2 = t, t 4 , 1, t 3 = t 4 , up to the indeterminacy. Tables 3 and 4 in Miller [9] (there is a typo in Table 4: the (4, 0) entry should start at 4 and not 5). The computer found no non-zero solutions of (10), implying that the deleted squares of L(11, 2) and L(11, 3) are not homotopy equivalent.…”

“…This question was studied by Miller [9] for two-point configuration spaces, also known under the name of deleted squares. Miller extended the Massey product calculation of [8] to arbitrary 2000 Mathematics Subject Classification.…”