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The evaluation subgroup of a fibre inclusion. (English) Zbl 1121.55011

Let \(f:X\to Y\) be a based map. The \(n\)-th evaluation subgroup \(G_n(Y,X; f)\) of \(f\) is the subgroup of \(\pi_n (Y)\) of elements represented by maps \(g:S^n\to Y\) such that \((g,f):S^n\vee X\to Y\) extends to \(S^n\times X\to Y\). The group \(G_n(X)=G_n(X,X; id_X)\) is the \(n\)-th Gottlieb group. It is the image of the map induced on homotopy groups by the evaluation map \(\omega: aut_1(X)\to X\), where \(aut_1(X)\) is the space of maps \(X\to X\) homotopic to \(id_X\). If \(X\) is a \(CW\)-complex of finite type, \(G_\ast(X)\) is known to be the image of the connecting homomorphism in the exact homotopy sequence of the universal fibration \(X\to U(X)\to B\, aut_1(X)\). Restricting the homotopy sequence of a fibration \(\xi:X{\overset {j} {} }E{\overset \varphi {} }B\) of simply connected \(CW\)-complexes of finite type we obtain the Gottlieb sequence \[ \cdots \to \pi_{n-1}(B){\overset \partial {} }G_n(X) {\overset j_\ast {} } G_n(E,X;j){\overset \varphi_\ast {} }\pi_n(B)\to\cdots \] because \(\partial: \pi_{n+1}(B)\to \pi_n(X)\) factors as \[ \pi_{n+1}(B){\overset h_\ast {} }\pi_{n+1}(B\,aut_1X){\overset \partial_\infty {} }\pi_n(X) \] where \(h:B\to B\,aut_1(X)\) is the classifying map and \(\partial_\infty\) the connecting homomorphism of the universal fibration. The Gottlieb sequence need not be exact, and we call \(\xi\) Gottlieb trivial if the sequence breaks into short exact sequences \[ 0\to G_n(X) {\overset j_\ast {} } G_n(E, X; j){\overset p_\ast {} }\pi_n(B)\to 0 \] for all \(n\geq 2\). Recall that \(\xi\) is fibre-homotopically trivial if \(\xi\) is fibre-homotopy equivalent to a trivial fibration and weak-homotopically trivial if \(\partial=0: \pi_{n+1}(B)\to\pi_n(X)\) for all \(n\). The authors prove the implications \[ \text{fibre-homotopically trivial} \Rightarrow \text{Gottlieb trivial} \Rightarrow \text{weak-homotopically trivial} \] and show that each of the reverse implications fails. Moreover, they work a number of examples and, using Sullivan’s minimal models, they give necessary and sufficient conditions for \(\xi\) to be rationally Gottlieb trivial, provided \(X\) is a finite \(CW\)-complex.

MSC:

55R15 Classification of fiber spaces or bundles in algebraic topology
55P62 Rational homotopy theory
55Q70 Homotopy groups of special types
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