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Fiber-homotopy self-equivalences and a classification of fibrations in rational homotopy

Research paper by Toshihiro Yamaguchi, Shoji Yokura

Indexed on: 28 Oct '16Published on: 17 Oct '16Published in: Journal of Homotopy and Related Structures



Abstract

Abstract For a fibration \(\xi : X \overset{i}{\rightarrow } E \overset{p}{\rightarrow } B\) we consider the image of the rationalized homotopy group homomorphism \(\pi _*(\mathrm{res} \ \xi )_{\mathbb Q}:\pi _*(\mathrm{aut}_1\mathrm{p})_{\mathbb Q} \rightarrow \pi _*(\mathrm{aut}_1\mathrm{X})_{\mathbb Q}\) obtained from the fibre-restricting map \(\mathrm{res}\ \xi : \mathrm{aut}_1\mathrm{p} \rightarrow \mathrm{aut}_1\mathrm{X}\) . Then we consider a finite type classification of fibrations \(\xi \) with fibre X and base B. In particular, we measure the size of it by a rational homotopical invariant “ \(\mathrm{depth}_B\mathrm{X}\) ” when X are certain homogeneous spaces and B are spheres.AbstractFor a fibration \(\xi : X \overset{i}{\rightarrow } E \overset{p}{\rightarrow } B\) we consider the image of the rationalized homotopy group homomorphism \(\pi _*(\mathrm{res} \ \xi )_{\mathbb Q}:\pi _*(\mathrm{aut}_1\mathrm{p})_{\mathbb Q} \rightarrow \pi _*(\mathrm{aut}_1\mathrm{X})_{\mathbb Q}\) obtained from the fibre-restricting map \(\mathrm{res}\ \xi : \mathrm{aut}_1\mathrm{p} \rightarrow \mathrm{aut}_1\mathrm{X}\) . Then we consider a finite type classification of fibrations \(\xi \) with fibre X and base B. In particular, we measure the size of it by a rational homotopical invariant “ \(\mathrm{depth}_B\mathrm{X}\) ” when X are certain homogeneous spaces and B are spheres. \(\xi : X \overset{i}{\rightarrow } E \overset{p}{\rightarrow } B\) \(\xi : X \overset{i}{\rightarrow } E \overset{p}{\rightarrow } B\) \(\pi _*(\mathrm{res} \ \xi )_{\mathbb Q}:\pi _*(\mathrm{aut}_1\mathrm{p})_{\mathbb Q} \rightarrow \pi _*(\mathrm{aut}_1\mathrm{X})_{\mathbb Q}\) \(\pi _*(\mathrm{res} \ \xi )_{\mathbb Q}:\pi _*(\mathrm{aut}_1\mathrm{p})_{\mathbb Q} \rightarrow \pi _*(\mathrm{aut}_1\mathrm{X})_{\mathbb Q}\) \(\mathrm{res}\ \xi : \mathrm{aut}_1\mathrm{p} \rightarrow \mathrm{aut}_1\mathrm{X}\) \(\mathrm{res}\ \xi : \mathrm{aut}_1\mathrm{p} \rightarrow \mathrm{aut}_1\mathrm{X}\) \(\xi \) \(\xi \)XB \(\mathrm{depth}_B\mathrm{X}\) \(\mathrm{depth}_B\mathrm{X}\)XB