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nLab > Latest Changes: rational parametrized homotopy theory

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeAug 29th 2020
- PermaLink
Author: Urs
Format: MarkdownItexstarting something... <a href="https://ncatlab.org/nlab/revision/rational+parametrized+homotopy+theory/1">v1</a>, <a href="https://ncatlab.org/nlab/show/rational+parametrized+homotopy+theory">current</a>

starting something…

v1, current
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeAug 29th 2020
- (edited Aug 30th 2020)
- PermaLink
Author: Urs
Format: MarkdownItexadded this statement: *** Let $$ \array{ F &\overset{fib}{\longrightarrow}& A \\ && \big\downarrow{}^{\mathrlap{p}} \\ && B } $$ be a [[Serre fibration]] of [[connected topological space|connected]] [[topological spaces]], with [[fiber]] $F$ (over any base point) also connected. If in addition 1. $\pi_1(B)$ [[nilpotent module|acts nilpotently]] on $H_\bullet(F,k)$ (e.g. if $B$ is [[simply connected topological space|simply connected]]); 1. at least one of $A$, $F$ have rational [[finite type]] then the [[cofiber]] of any [[relative Sullivan model]] for $p$ is a [[Sullivan model]] for $F$. *** <a href="https://ncatlab.org/nlab/revision/diff/rational+parametrized+homotopy+theory/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/rational+parametrized+homotopy+theory/2">v2</a>, <a href="https://ncatlab.org/nlab/show/rational+parametrized+homotopy+theory">current</a>
added this statement:

Let
Ffib⟶A↓pB

be a Serre fibration of connected topological spaces, with fiber $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>$ (over any base point) also connected.

If in addition
1. $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(B)</annotation></semantics></math>$ acts nilpotently on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_\bullet(F,k)</annotation></semantics></math>$
  
  (e.g. if $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>$ is simply connected);
2. at least one of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>$ , $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>$ have rational finite type
then the cofiber of any relative Sullivan model for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>$ is a Sullivan model for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>$ .

diff, v2, current
- CommentRowNumber3.
- CommentAuthorDavidRoberts
- CommentTimeAug 29th 2020
- PermaLink
Author: DavidRoberts
Format: MarkdownItexDid you mean $\pi_1(Y)$ acts nilpotently?

Did you mean $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(Y)</annotation></semantics></math>$ acts nilpotently?
- CommentRowNumber4.
- CommentAuthorUrs
- CommentTimeAug 30th 2020
- PermaLink
Author: Urs
Format: MarkdownItexYes, I had fixed it in the entry, didn't fix it here.

Yes, I had fixed it in the entry, didn’t fix it here.
- CommentRowNumber5.
- CommentAuthorUrs
- CommentTimeAug 30th 2020
- (edited Aug 30th 2020)
- PermaLink
Author: Urs
Format: MarkdownItexadded the addendum: *** Moreover, if $CE(\mathfrak{l}B)$ is a [[minimal Sullivan model]] for $B$, then the cofiber of the corresponding minimal [[relative Sullivan model]] for $p$ is the [[minimal Sullivan model]] $CE(\mathfrak{l}F)$ for $F$: $$ \array{ CE(\mathfrak{l}F) &\overset{ cofib \big( CE(\mathfrak{l}p) \big) }{\longleftarrow}& CE(\mathfrak{l}_{{}_B}A) \\ && \big\downarrow{}^{\mathrlap{ CE(\mathfrak{l}p) }} \\ && CE(\mathfrak{l}B) } $$ *** <a href="https://ncatlab.org/nlab/revision/diff/rational+parametrized+homotopy+theory/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/rational+parametrized+homotopy+theory/3">v3</a>, <a href="https://ncatlab.org/nlab/show/rational+parametrized+homotopy+theory">current</a>

added the addendum:

Moreover, if $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔩</mi><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\mathfrak{l}B)</annotation></semantics></math>$ is a minimal Sullivan model for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>$ , then the cofiber of the corresponding minimal relative Sullivan model for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>$ is the minimal Sullivan model $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔩</mi><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\mathfrak{l}F)</annotation></semantics></math>$ for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>$ :
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtable><mtr><mtd><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔩</mi><mi>F</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟵</mo><mrow><mi>cofib</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔩</mi><mi>p</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mover></mtd> <mtd><mi>CE</mi><mo stretchy="false">(</mo><msub><mi>𝔩</mi> <mrow><msub><mrow/> <mi>B</mi></msub></mrow></msub><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd/> <mtd/> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow/> <mpadded width="0px"><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔩</mi><mi>p</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd/> <mtd/> <mtd><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔩</mi><mi>B</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ CE(\mathfrak{l}F) &amp;\overset{ cofib \big( CE(\mathfrak{l}p) \big) }{\longleftarrow}&amp; CE(\mathfrak{l}_{{}_B}A) \\ &amp;&amp; \big\downarrow{}^{\mathrlap{ CE(\mathfrak{l}p) }} \\ &amp;&amp; CE(\mathfrak{l}B) } </annotation></semantics></math>$

diff, v3, current
- CommentRowNumber6.
- CommentAuthorUrs
- CommentTimeAug 31st 2020
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to: * [[Aldridge Bousfield]], [[Daniel Kan]], Chapter II "Fiber Lemmas" of: _Homotopy Limits, Completions and Localizations_, Springer 1972 ([doi:10.1007/978-3-540-38117-4](https://doi.org/10.1007/978-3-540-38117-4)) <a href="https://ncatlab.org/nlab/revision/diff/rational+parametrized+homotopy+theory/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/rational+parametrized+homotopy+theory/4">v4</a>, <a href="https://ncatlab.org/nlab/show/rational+parametrized+homotopy+theory">current</a>
added pointer to:
- Aldridge Bousfield, Daniel Kan, Chapter II “Fiber Lemmas” of: Homotopy Limits, Completions and Localizations, Springer 1972 (doi:10.1007/978-3-540-38117-4)
diff, v4, current
- CommentRowNumber7.
- CommentAuthorUrs
- CommentTimeAug 31st 2020
- PermaLink
Author: Urs
Format: MarkdownItexadded the following further addendum to the proposition: *** But this cofiber, being the cofiber of a [[relative Sullivan model]] and hence of a [[cofibration]] in the [[projective model structure on dgc-algebras]], is in fact the [[homotopy cofiber]], and hence is a model for the [[homotopy fiber]] of the [[rationalization|rationalized]] fibration. Therefore (eq:SullivanModelForFiber) implies that on fibrations of connected finite-type spaces where $\pi_1$ of the base acts nilpotently on the homology of the fiber, [[rationalization]] preserves [[homotopy fibers]]. (This was originally proven in [Bousfield-Kan 72, Chapter II](#BousfieldKan72).) *** <a href="https://ncatlab.org/nlab/revision/diff/rational+parametrized+homotopy+theory/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/rational+parametrized+homotopy+theory/4">v4</a>, <a href="https://ncatlab.org/nlab/show/rational+parametrized+homotopy+theory">current</a>

added the following further addendum to the proposition:

But this cofiber, being the cofiber of a relative Sullivan model and hence of a cofibration in the projective model structure on dgc-algebras, is in fact the homotopy cofiber, and hence is a model for the homotopy fiber of the rationalized fibration.

Therefore (eq:SullivanModelForFiber) implies that on fibrations of connected finite-type spaces where $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\pi_1</annotation></semantics></math>$ of the base acts nilpotently on the homology of the fiber, rationalization preserves homotopy fibers.

(This was originally proven in Bousfield-Kan 72, Chapter II.)

diff, v4, current
- CommentRowNumber8.
- CommentAuthorUrs
- CommentTimeSep 3rd 2020
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer also to the followup book * [[Yves Félix]], [[Steve Halperin]], [[Jean-Claude Thomas]], Sections 4 and 5 of: _Rational Homotopy Theory II_, World Scientific 2015 ([doi:10.1142/9473](https://doi.org/10.1142/9473)) where the above statement appears as Theorem 5.1 <a href="https://ncatlab.org/nlab/revision/diff/rational+parametrized+homotopy+theory/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/rational+parametrized+homotopy+theory/5">v5</a>, <a href="https://ncatlab.org/nlab/show/rational+parametrized+homotopy+theory">current</a>
added pointer also to the followup book
- Yves Félix, Steve Halperin, Jean-Claude Thomas, Sections 4 and 5 of: Rational Homotopy Theory II, World Scientific 2015 (doi:10.1142/9473)
where the above statement appears as Theorem 5.1

diff, v5, current

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nLab > Latest Changes: rational parametrized homotopy theory