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nLab > Latest Changes: integral Steenrod square

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeFeb 19th 2018
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Author: Urs
Format: MarkdownItexstarted a minimum for _[[integral Steenrod square]]_

started a minimum for integral Steenrod square
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeFeb 19th 2018
- PermaLink
Author: Urs
Format: MarkdownItexI have added as an example [here](https://ncatlab.org/nlab/show/integral+Steenrod+square#IntegralSteenrodSquareRefinedToOrdinaryDifferentialCohomology) the observation that after canonical lifting $\widehat{Sq}^{2n+1}_{\mathbb{Z}}$ of the integral Steenrod squares to ordinary differential cohomology and then restriction to the subspace of cocyles whose underlying integral Steenrod square vanishes, we obtain a canonical map $$ H^{2n+2}_{diff}(X)|_{Sq^{2n+1}_{\mathbb{Z}} = 0} \overset{\widehat{Sq}_{\mathbb{Z}}^{2n+1}}{\longrightarrow} H^{4n+3}_{dR}(X) $$ from differential cohomology (subject to that condition) in degree $2n+2$ to de Rham cohomology in degree $3n+3$. I'd like to have an explicit description of this map on differential form representatives...

I have added as an example here the observation that after canonical lifting $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mover><mi>Sq</mi><mo>^</mo></mover> <mi>ℤ</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">\widehat{Sq}^{2n+1}_{\mathbb{Z}}</annotation></semantics></math>$ of the integral Steenrod squares to ordinary differential cohomology and then restriction to the subspace of cocyles whose underlying integral Steenrod square vanishes, we obtain a canonical map
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>H</mi> <mi>diff</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mrow><msubsup><mi>Sq</mi> <mi>ℤ</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>=</mo><mn>0</mn></mrow></msub><mover><mo>⟶</mo><mrow><msubsup><mover><mi>Sq</mi><mo>^</mo></mover> <mi>ℤ</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mrow></mover><msubsup><mi>H</mi> <mi>dR</mi> <mrow><mn>4</mn><mi>n</mi><mo>+</mo><mn>3</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^{2n+2}_{diff}(X)|_{Sq^{2n+1}_{\mathbb{Z}} = 0} \overset{\widehat{Sq}_{\mathbb{Z}}^{2n+1}}{\longrightarrow} H^{4n+3}_{dR}(X) </annotation></semantics></math>$
from differential cohomology (subject to that condition) in degree $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">2n+2</annotation></semantics></math>$ to de Rham cohomology in degree $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><mi>n</mi><mo>+</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">3n+3</annotation></semantics></math>$ .

I’d like to have an explicit description of this map on differential form representatives…
- CommentRowNumber3.
- CommentAuthornLab edit announcer
- CommentTimeNov 19th 2020
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Author: nLab edit announcer
Format: MarkdownItexfixed grading (Sq^2n adds +2n) Anonymous <a href="https://ncatlab.org/nlab/revision/diff/integral+Steenrod+square/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/integral+Steenrod+square/4">v4</a>, <a href="https://ncatlab.org/nlab/show/integral+Steenrod+square">current</a>

fixed grading (Sq^2n adds +2n)

Anonymous

diff, v4, current

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nLab > Latest Changes: integral Steenrod square