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A fibre-restricted Gottlieb group and its rational realization problem. (English) Zbl 1334.55004

Let \(p: E\to Y\) be a fibration with fibre \(X= p^{-1}(\{y_0\})\) for some fixed \(y_0\in Y\). The monoid of (free) fibre homotopy equivalences of \(p\), which are (freely) homotopic to the identity map, is denoted \(\text{aut}_1(p)\). If \(Y= \{y_0\}\) then we recover the component \(\text{aut}_1(X)\) of the identity map in the usual monoid of self-equivalences \(\text{aut}(X)\). Since a fibre homotopy equivalence of \(p\) induces a homotopy equivalence of \(X\) there is a map \(R: \text{aut}_1(p)\to\text{aut}_1(X)\) and, for a fixed \(x_0\in X\), we have the evaluation map \(ev: \text{aut}_1(X)\to X\). Consider \(\text{aut}_1(X)\) and \(\text{aut}_1(p)\subset\text{aut}_1(E)\) as topological spaces with the compact-open topology and pointed by the identity map. The image of the composite \(\pi_n(ev\circ R)\) is a subgroup of \(\pi_n(X, x_0)\), denoted \(G_n(p)\). If \(Y= \{y_0\}\) then we recover the usual \(n\)th-Gottlieb group \(G_n(X)\) and, essentially by definition, \(G_n(p)\subset G_n(X)\).
The author studies the poset, under inclusion, of the \(\mathbb{Q}\)-vector spaces \(G_n(p)\) when \(p: E\to Y\) is a fibration admitting \(X\) as a fiber such that \(X\), \(Y\) are 1-connected, \(X\), \(E\), \(Y\) are rational spaces. He proves, using Sullivan rational homotopy theory, some results concerning descending chains \(G_n(X)\supset G_n(p_1)\supset G_n(p_2)\cdots\) when \(X\) is an elliptic space.

MSC:

55P62 Rational homotopy theory
55Q52 Homotopy groups of special spaces
55P10 Homotopy equivalences in algebraic topology
55R10 Fiber bundles in algebraic topology
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