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Relative phantom maps and rational homotopy. (English) Zbl 1475.55016

Let \(X\) be a CW-complex of finite type. A map \(X\rightarrow Y\) is called a phantom map if its restriction to each skeleton of \(X\) is null-homotopic. Recently, K. Iriye, D. Kishimoto and T. Matsushita [K. Iriye et al., Algebr. Geom. Topol. 19, No. 1, 341–362 (2019; Zbl 1419.55013)] introduced a relative version of phantom maps defined as follows: Let \(X\) be a CW-complex of finite type and \(\varphi:B\rightarrow Y\) be a map between spaces. A map \(X\rightarrow Y\) is called a relative phantom map from \(X\) to \(\varphi\) if its restriction to each skeleton of \(X\) lifts to \(B\) through \(\varphi\), up to homotopy.
In this paper, the authors generalize some results of B. J. Gray [“Operations and a problem of Heller”, PhD-Thesis, University of Chicago (1965)] and C. A. McGibbon and J. Roitberg [Am. J. Math. 116, No. 6, 1365–1379 (1994; Zbl 0842.55008)] on relations between phantom maps and rational homotopy to relative phantom maps. In particular, they show the following results:
Proposition. Given a map \(\varphi:B\rightarrow Y\), suppose that \(\pi_{1}(B)\) acts trivially on \(\pi_{\ast}(Y,B)\). If there is a non-trivial relative phantom map from \(X\) to \(\varphi\), then \(H^{n}(X;\mathbb{Q})\neq0\) and \(\pi_{n+1}(Y,B)\otimes\mathbb{Q}\neq0\) for some integer \(n\geq1\).
Theorem. Let \(X\) be a finite type source, \(B\), \(Y\) and \(Y^{\prime}\) be simply connected finite targets and \(\varphi:B\rightarrow Y\) be a map. If \(X\) is a suspension and \(f:Y\rightarrow Y^{\prime}\) induces a surjection \(f_{\ast} :\pi_{\ast}(Y)\otimes \mathbb{Q} \rightarrow\pi_{\ast}(Y^{\prime})\otimes \mathbb{Q}\), then \[ f_{\ast}:Ph(X,\varphi)\rightarrow Ph(X,f\circ \varphi) \] is surjective.
The paper ends with some problems on relative phantom maps and rational homotopy.

MSC:

55P99 Homotopy theory
55P62 Rational homotopy theory
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References:

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