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Consider any fibration
p : E → B, any finite C. W.—pair (K,L), and any maps f : K → B and h : L → E
such that p ∘ h = f|L. A map g : K → E such that p ∘ g = f and g|L = h we call a
lifting of frelh.
In this paper single obstruction Γ(f) ∈ H′(K,L,f;g) is defined. g is a so-called
B-spectrum, and H∗ ( ; g) is cohomology in that spectrum. If a lifting of f rel h
exists, Γ(f) = 0; this condition is also sufficient if the fiber of p is k-connected and
dim(K∕L) ≦ 2k + 1.
If g0 and g1 are liftings of f rel h, a single obstruction δ(g0,g1;h) ∈ H(K,L,f : g)
is also defined; if g0 and g1 are connected by a homotopy of liftings of f rel
hδ(g0,g1;h) = 0; this condition is, also sufficient if p is k-connected and
dim(K∕L) ≦ 2k.
In §4, a spectral sequence is constructed for cohomology in a B-spectrum, based
on the Postnikov tower of that spectrum, and the relationship between the single
obstruction and the classical obstructions is defined.
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