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ort96
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$\require{AMScd}$ For rational fibrations $F \rightarrow E \xrightarrow{\pi} B$$F \rightarrow E \rightarrow B$, we have a nice description in terms of sullivan minimal models, namely a commutative diagram of cdga's $$ \begin{CD} A_{PL}(B) @>>> A_{PL}(E) @>{\pi}>> A_{PL}(F) \\ @AAA @AAA @AAA \\ (\wedge V ,d_{B}) @>>> (\wedge V \otimes \wedge Z,d) @>>> (\wedge Z,d_{F}) \end{CD} $$$$ \begin{CD} A_{PL}(B) @>>> A_{PL}(E) @>>> A_{PL}(F) \\ @AAA @AAA @AAA \\ (\wedge V ,d_{B}) @>>> (\wedge V \otimes \wedge Z,d) @>>> (\wedge Z,d_{F}) \end{CD} $$ where all vertical arrows are minimal models. This is sometimes called a $\wedge$-minimal $\wedge$-model. See for instance https://www.jstor.org/stable/pdf/1997895.pdf?refreqid=excelsior%3Aacbce1fbb0022d42e2fe7d1265efcb8c

Is this construction natural in some sense? Say for 'morphisms of fibre bundles' or 'morphisms of fibre bundles preserving a common $G$-structure'.

$\require{AMScd}$ For rational fibrations $F \rightarrow E \xrightarrow{\pi} B$, we have a nice description in terms of sullivan minimal models, namely a commutative diagram of cdga's $$ \begin{CD} A_{PL}(B) @>>> A_{PL}(E) @>{\pi}>> A_{PL}(F) \\ @AAA @AAA @AAA \\ (\wedge V ,d_{B}) @>>> (\wedge V \otimes \wedge Z,d) @>>> (\wedge Z,d_{F}) \end{CD} $$ where all vertical arrows are minimal models. This is sometimes called a $\wedge$-minimal $\wedge$-model. See for instance https://www.jstor.org/stable/pdf/1997895.pdf?refreqid=excelsior%3Aacbce1fbb0022d42e2fe7d1265efcb8c

Is this construction natural in some sense? Say for 'morphisms of fibre bundles' or 'morphisms of fibre bundles preserving a common $G$-structure'.

$\require{AMScd}$ For rational fibrations $F \rightarrow E \rightarrow B$, we have a nice description in terms of sullivan minimal models, namely a commutative diagram of cdga's $$ \begin{CD} A_{PL}(B) @>>> A_{PL}(E) @>>> A_{PL}(F) \\ @AAA @AAA @AAA \\ (\wedge V ,d_{B}) @>>> (\wedge V \otimes \wedge Z,d) @>>> (\wedge Z,d_{F}) \end{CD} $$ where all vertical arrows are minimal models. This is sometimes called a $\wedge$-minimal $\wedge$-model. See for instance https://www.jstor.org/stable/pdf/1997895.pdf?refreqid=excelsior%3Aacbce1fbb0022d42e2fe7d1265efcb8c

Is this construction natural in some sense? Say for 'morphisms of fibre bundles' or 'morphisms of fibre bundles preserving a common $G$-structure'.

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ort96
ort96
  • 404
  • 2
  • 7

$\require{AMScd}$ For a rational fibrations $F \rightarrow E \xrightarrow{\pi} B$, we have a nice description in terms of sullivan minimal models, namely a commutative diagram of cdga's $$ \begin{CD} A_{PL}(B) @>>> A_{PL}(E) @>{\pi}>> A_{PL}(F) \\ @AAA @AAA @AAA \\ (\wedge V ,d_{B}) @>>> (\wedge V \otimes \wedge Z,d) @>>> (\wedge Z,d_{F}) \end{CD} $$ where all vertical arrows are minimal models. This is sometimes called a $\wedge$-minimal $\wedge$-model. See for instance https://www.jstor.org/stable/pdf/1997895.pdf?refreqid=excelsior%3Aacbce1fbb0022d42e2fe7d1265efcb8c

Is this construction natural in some sense? Say for 'morphisms of fibre bundles' or 'morphisms of fibre bundles preserving a common $G$-structure'.

$\require{AMScd}$ For a rational fibrations $F \rightarrow E \xrightarrow{\pi} B$, we have a nice description in terms of sullivan minimal models, namely a commutative diagram of cdga's $$ \begin{CD} A_{PL}(B) @>>> A_{PL}(E) @>{\pi}>> A_{PL}(F) \\ @AAA @AAA @AAA \\ (\wedge V ,d_{B}) @>>> (\wedge V \otimes \wedge Z,d) @>>> (\wedge Z,d_{F}) \end{CD} $$ where all vertical arrows are minimal models. This is sometimes called a $\wedge$-minimal $\wedge$-model. See for instance https://www.jstor.org/stable/pdf/1997895.pdf?refreqid=excelsior%3Aacbce1fbb0022d42e2fe7d1265efcb8c

Is this construction natural in some sense? Say for 'morphisms of fibre bundles' or 'morphisms of fibre bundles preserving a common $G$-structure'.

$\require{AMScd}$ For rational fibrations $F \rightarrow E \xrightarrow{\pi} B$, we have a nice description in terms of sullivan minimal models, namely a commutative diagram of cdga's $$ \begin{CD} A_{PL}(B) @>>> A_{PL}(E) @>{\pi}>> A_{PL}(F) \\ @AAA @AAA @AAA \\ (\wedge V ,d_{B}) @>>> (\wedge V \otimes \wedge Z,d) @>>> (\wedge Z,d_{F}) \end{CD} $$ where all vertical arrows are minimal models. This is sometimes called a $\wedge$-minimal $\wedge$-model. See for instance https://www.jstor.org/stable/pdf/1997895.pdf?refreqid=excelsior%3Aacbce1fbb0022d42e2fe7d1265efcb8c

Is this construction natural in some sense? Say for 'morphisms of fibre bundles' or 'morphisms of fibre bundles preserving a common $G$-structure'.

Source Link
ort96
ort96
  • 404
  • 2
  • 7

Naturality of minimal model of a fibre bundle

$\require{AMScd}$ For a rational fibrations $F \rightarrow E \xrightarrow{\pi} B$, we have a nice description in terms of sullivan minimal models, namely a commutative diagram of cdga's $$ \begin{CD} A_{PL}(B) @>>> A_{PL}(E) @>{\pi}>> A_{PL}(F) \\ @AAA @AAA @AAA \\ (\wedge V ,d_{B}) @>>> (\wedge V \otimes \wedge Z,d) @>>> (\wedge Z,d_{F}) \end{CD} $$ where all vertical arrows are minimal models. This is sometimes called a $\wedge$-minimal $\wedge$-model. See for instance https://www.jstor.org/stable/pdf/1997895.pdf?refreqid=excelsior%3Aacbce1fbb0022d42e2fe7d1265efcb8c

Is this construction natural in some sense? Say for 'morphisms of fibre bundles' or 'morphisms of fibre bundles preserving a common $G$-structure'.