Vol. 41, No. 3, 1972

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Twisted cohomology theories and the single obstruction to lifting

Lawrence Louis Larmore

Vol. 41 (1972), No. 3, 755–769

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 (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.

Mathematical Subject Classification
Primary: 55G35
Received: 23 June 1970
Revised: 8 April 1971
Published: 1 June 1972
Lawrence Louis Larmore