Cohomology ${\rm mod} p$ of the $p$ -fold symmetric products of spheres.

Remarks on motivic homotopy theory over algebraically closed fields

“…Now we see that an A 1 -homotopy inverse for the map in the assumption of Lemma 10 is in our case obtained simply by 1-point compactifying (18), (20) in the A nk − ∆ coordinate. We obtain Proposition 12.…”

“…We are essentially interested in mimicking the computation of the cohomology of symmetric products of spheres by Nakaoka [20] in the motivic situation. We claim that condition (13) is satisfied when we set (10) with Σ d acting on d by the standard permutation representation.…”

Realizing transfer maps for ramified coverings

“…Now if Y is any space consider the projection h n \ Y n X Σn EΣ n -> Y n /Σ n = SP" (7). Observe that if p: X/H -> X/G is the projection of orbit spaces coming from a G-action on X, then the following diagram commutes:…”

“…Let The reference for these homology structures is [7]. A quick examination of these diagrams yields that the transfer homomorphisms do not preserve the A ^-structures.…”

Symmetric Joins and Weighted Barycenters

“…We indicate below a streamlined construction of a transfer map which appeals as before to the identification of ΣB n (M ) with SP n (ΣM ). In our case and for closed oriented M we seek to construct for each positive n a map (25) Θ : H n(d+1) (SP n (ΣM ))− −− →H (n−1)(d+1) (SP n−1 (ΣM )) with the right homological properties. The construction is due to L. Smith and has refinements in [9].…”