Determine the homotopy type of the third filtration of the relative James construction for general pairs (M_f, X)

Determine the homotopy type of the third filtration J3(M_f, X) of the relative James construction J(M_f, X) for a general continuous map f: X → Y between CW complexes, where M_f denotes the mapping cylinder of f and X ↪ M_f is the inclusion. Specifically, establish a homotopy-equivalent CW description (including the attaching maps) of J3(M_f, X) beyond the special Moore space cases in which this filtration is already known.

Background

Gray’s relative James construction J(X, A) for a CW-pair A ↪ X is filtered by subspaces J_r(X, A). For a map f: X → Y with mapping cylinder M_f, the space J(M_f, X) is homotopy equivalent to the homotopy fiber of the pinch map on the mapping cone, making its filtrations central to computations of homotopy groups beyond the metastable range.

While the second filtration J2(X, A) has a known description as a cofiber with attaching map given by a generalized Whitehead product, the homotopy type of the third filtration J3(M_f, X) is generally not determined. It is known in certain special cases (e.g., mod 2 Moore spaces with k = 3,5), but a general characterization remains unavailable. The paper provides new structural information by identifying that the attaching maps for Jr(X, A) lie in sets of higher order Whitehead products, but this does not by itself specify the exact homotopy type of J3(M_f, X) in general.