Induced stratification

Induced Stratification - Stratification $f^{*}(\pi )\colon E'\to B'$ ${\ displaystyle f ^ {*} (\ pi) \ colon E '\ to B'}$ ${\ displaystyle f ^ {*} (\ pi) \ colon E '\ to B'}$ induced by the mapping $f\colon B'\to B$ ${\ displaystyle f \ colon B '\ to B}$ ${\ displaystyle f \ colon B '\ to B}$ and bundle $\pi \colon E\to B$ ${\ displaystyle \ pi \ colon E \ to B}$ ${\ displaystyle \ pi \ colon E \ to B}$ where $E^{'}$ ${\ displaystyle E '}$ $E '$ - subspace of direct product $B'\times E$ ${\ displaystyle B '\ times E}$ ${\ displaystyle B '\ times E}$ consisting of pairs $(b',e)$ ${\ displaystyle (b ', e)}$ ${\ displaystyle (b ', e)}$ for which $f(b')=\pi (e)$ ${\ displaystyle f (b ') = \ pi (e)}$ ${\ displaystyle f (b ') = \ pi (e)}$ , and $f^{*}(\pi )\colon (b',e)\mapsto b'$ ${\ displaystyle f ^ {*} (\ pi) \ colon (b ', e) \ mapsto b'}$ ${\ displaystyle f ^ {*} (\ pi) \ colon (b ', e) \ mapsto b'}$ .

In this case, the following commutative diagram forms a Cartesian square :

Properties

Display $F$ $:$ $E$ $'$ $\to$ $E$ {\ displaystyle F \ colon E '\ to E} induced bundle into the original bundle defined by the formula $F$ $($ $b$ $'$ $,$ $e$ $)$ $=$ $e$ {\ displaystyle F (b ', e) = e} is a bundle morphism covering $f$ {\ displaystyle f} .
- For every point $b'\in B'$ ${\ displaystyle b '\ in B'}$ restrictions on the layer is homeomorphism.
For any bundle $\eta \colon X\to B'$ ${\ displaystyle \ eta \ colon X \ to B '}$ and morphism $H:\eta \to \pi$ ${\ displaystyle H: \ eta \ to \ pi}$ covering $f$ ${\ displaystyle f}$ , there is one and only one morphism $K:\eta \to f^{*}(\pi )$ ${\ displaystyle K: \ eta \ to f ^ {*} (\ pi)}$ satisfying the relations
$FK=H$ ${\ displaystyle FK = H}$
$f*(\pi )K=\eta$ ${\ displaystyle f * (\ pi) K = \ eta}$ .
The bundles induced by isomorphic bundles are isomorphic, the bundle induced by a constant map is isomorphic to the trivial one.
For any section $s$ ${\ displaystyle s}$ bundles $\pi$ ${\ displaystyle \ pi}$ display $\sigma \colon B'\to E'$ ${\ displaystyle \ sigma \ colon B '\ to E'}$ defined by the formula $\sigma (b')=(b',sf(b'))$ ${\ displaystyle \ sigma (b ') = (b', sf (b '))}$ is a section of the induced bundle $f^{*}(\pi )$ ${\ displaystyle f ^ {*} (\ pi)}$ and satisfies the relation $F\sigma =sf$ ${\ displaystyle F \ sigma = sf}$ .

Induced stratification

Properties

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