Индуцированное расслоение

Индуцированное расслоение — расслоение $f^{*}(\pi )\colon E'\to B'$ , индуцированное отображением $f\colon B'\to B$ и расслоением $\pi \colon E\to B$ , где $E'$ — подпространство прямого произведения $B'\times E$ , состоящее из пар $(b',e)$ , для которых $f(b')=\pi (e)$ , и $f^{*}(\pi )\colon (b',e)\mapsto b'$ .

При этом следующая коммутативная диаграмма образует декартов квадрат:

Свойства

Отображение $F\colon E'\to E$ $F\colon E'\to E$ индуцированного расслоения в исходное расслоение, определённое формулой $F(b',e)=e$ $F(b',e)=e$ , является морфизмом расслоений, накрывающим $f$ $f$ .
- Для каждой точки $b'\in B'$ ограничения на слой является гомеоморфизмами.
Для любого расслоения $\eta \colon X\to B'$ и морфизма $H:\eta \to \pi$ , накрывающего $f$ , существует один и только один морфизм $K:\eta \to f^{*}(\pi )$ , удовлетворяющий соотношениям
$FK=H$

$f*(\pi )K=\eta$ .
Расслоения, индуцированные изоморфными расслоениями, изоморфны, расслоение, индуцированное постоянным отображением, изоморфно тривиальному.
Для любого сечения $s$ расслоения $\pi$ отображение $\sigma \colon B'\to E'$ , определённое формулой $\sigma (b')=(b',sf(b'))$ , является сечением индуцированного расслоения $f^{*}(\pi )$ и удовлетворяет соотношению $F\sigma =sf$ .

Литература

Хатчер А. Алгебраическая топология. — М.: МЦНМО, 2011. — 688 с. — ISBN 978-5-94057-748-5.

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