On a DGL-map between derivations of Sullivan minimal models. (English) Zbl 1459.55007
Given a simply connected CW-complex \(X\) of finite type, let aut\(_{1}\left(X\right) =\) map\(\left( X,X;id\right) \) denote the identity component of the space aut\(\left( X\right) \) of self-equivalences of \(X\). The group-like space aut\(_{1}\left( X\right) \) has a classifying space Baut\(_{1}\left( X\right) \). It appears as the base space of the universal fibration \(X\rightarrow UE\rightarrow\)Baut\(_{1}\left( X\right) \).
Let \(f:X\rightarrow Y\) be a fibration of simply connected finite CW-complexs. In this paper, the author gives partial answers to the following question: When does the map \(f\) strictly induce a map \(a_{f}:\left( \text{Baut}_{1}\left(X\right)\right) _{\mathbb{Q}}\rightarrow\left( \text{Baut}_{1}\left( Y\right) \right)_{\mathbb{Q}}?\)
Some results for this question are obtained by Theorem \(1.5\). In particular, a rationally weakly trivial map \(f:X\rightarrow Y\) strictly induces \(a_{f}:\left( \text{Baut}_{1}\left( X\right) \right) _{\mathbb{Q}}\rightarrow\left(\text{Baut}_{1}\left(Y\right)\right)_{\mathbb{Q}}\) if and only if it is \(\pi_{\mathbb{Q}}\)-separable.
In Section \(3\), the author gives some conditions that the strictly induced map \(a_{f}:\left(\text{Baut}_{1}\left(X\right)\right) _{\mathbb{Q}}\rightarrow\left(\text{Baut}_{1}\left(Y\right)\right)_{\mathbb{Q}}\) admits a section.
The paper finishes with many other interesting results (see Section \(4\) and also \(5\)).
Let \(f:X\rightarrow Y\) be a fibration of simply connected finite CW-complexs. In this paper, the author gives partial answers to the following question: When does the map \(f\) strictly induce a map \(a_{f}:\left( \text{Baut}_{1}\left(X\right)\right) _{\mathbb{Q}}\rightarrow\left( \text{Baut}_{1}\left( Y\right) \right)_{\mathbb{Q}}?\)
Some results for this question are obtained by Theorem \(1.5\). In particular, a rationally weakly trivial map \(f:X\rightarrow Y\) strictly induces \(a_{f}:\left( \text{Baut}_{1}\left( X\right) \right) _{\mathbb{Q}}\rightarrow\left(\text{Baut}_{1}\left(Y\right)\right)_{\mathbb{Q}}\) if and only if it is \(\pi_{\mathbb{Q}}\)-separable.
In Section \(3\), the author gives some conditions that the strictly induced map \(a_{f}:\left(\text{Baut}_{1}\left(X\right)\right) _{\mathbb{Q}}\rightarrow\left(\text{Baut}_{1}\left(Y\right)\right)_{\mathbb{Q}}\) admits a section.
The paper finishes with many other interesting results (see Section \(4\) and also \(5\)).
Reviewer: Abdelhadi Zaim (Casablanca)
MSC:
| 55P62 | Rational homotopy theory |
| 55R15 | Classification of fiber spaces or bundles in algebraic topology |
Keywords:
Dold-Lashof classifying space; rational homotopy theory; Sullivan minimal model; Quillen model; derivation; Halperin conjectureReferences:
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