# 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