In the situation of the relative lifting problem, suppose one has two different lifts \(W \to X\) that agree on the subspace \(A \subset W\). Show that the obstructions to finding a homotopy rel \(A\) between these two lifts lie in the groups \(H^n(W, A; \pi_n F)\).
Solution
1. Reformulation on the cylinder \(W \times I\):
1.1. Let \(p: X \to B\) be a fibration with fiber \(F\). Suppose \(f_0, f_1: W \to X\) are two lifts of \(g: W \to B\) such that \(f_0|_A = f_1|_A\).
setup.
1.2. A homotopy \(H: W \times I \to X\) between \(f_0\) and \(f_1\) rel \(A\) lifting \(G(w, t) = g(w)\) is equivalent to extending a partially defined lift \(H\) from the subspace:\[ Y = (W \times \{0\}) \cup (A \times I) \cup (W \times \{1\}) \subset W \times I \]to the entire cylinder \(W \times I\), where \(H(w, 0) = f_0(w)\), \(H(w, 1) = f_1(w)\), and \(H(a, t) = f_0(a) = f_1(a)\) for \(a \in A\).
definition of relative homotopy between lifts.
2. Cellular decomposition of the cylinder pair:
2.1. Endow the CW pair \((W, A)\) with a relative CW structure.The relative product cell structure on \((W \times I, Y)\) has cells in bijective correspondence with the cells of \(W \setminus A\):An \(n\)-cell \(e^n\) of \(W \setminus A\) gives rise to an \((n+1)\)-cell \(e^n \times (0, 1)\) in \((W \times I) \setminus Y\).
product cellular decomposition with \(I = [0, 1]\).
2.2. The boundary of the cell \(e^n \times I\) is:\[ \partial(e^n \times I) = (e^n \times \{0\}) \cup (\partial e^n \times I) \cup (e^n \times \{1\}) \cong S^n. \]
boundary formula for cylinder cells.
3. Obstruction cochains and groups:
3.1. Suppose \(H\) has been extended to \((W \times I)^{n} \cup Y\).To extend \(H\) across an \((n+1)\)-cell \(e^n \times (0, 1)\), \(H\) is already defined on the boundary \(\partial(e^n \times I) \cong S^n\).Since \(p \circ H = g \circ \operatorname{pr}_W\) is contractible on \(e^n \times I\), the map on the boundary takes values in the fiber \(F\), determining an element of \(\pi_n(F)\).
standard obstruction theory for extending sections of fibrations.
3.2. This defines an obstruction cocycle:\[ c^{n+1} \in C^{n+1}(W \times I, Y; \pi_n F). \]
assignment of boundary homotopy classes to \((n+1)\)-cells.
3.3. By the suspension isomorphism for relative CW pairs:\[ C^{n+1}(W \times I, (W \times \partial I) \cup (A \times I); \pi_n F) \cong C^n(W, A; \pi_n F), \]which passes to cohomology to yield:\[ H^{n+1}(W \times I, Y; \pi_n F) \cong H^n(W, A; \pi_n F). \]
excision and suspension isomorphism in relative cohomology.
4. Conclusion:The obstruction to finding a homotopy rel \(A\) between the two lifts at stage \(n\) lies in \(H^n(W, A; \pi_n F)\). Q.E.D.
2 and 3.