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nLab > Latest Changes: cyclic loop space

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeFeb 14th 2017
- PermaLink
Author: Urs
Format: MarkdownItexI need a word for the homotopy quotient $(\mathcal{L}X)/S^1$ of free loop spaces $\mathcal{L}X$ by their canonical circle action. It seems that the only term in use with respect to this is "twisted loop space", which however usually refers just to the constant loops $(\mathcal{L}_{const}X)//S^1$. Since under nice conditions the derived functions on the $\mathcal{L}Spec(A)/S^1$ is the cyclic homology complex of $A$, I suggest that a good name is "cyclic loop space". I made a quick note at _[[cyclic loop space]]_, just to fix and disambiguate terminology.

I need a word for the homotopy quotient $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>ℒ</mi><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">(\mathcal{L}X)/S^1</annotation></semantics></math>$ of free loop spaces $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X</annotation></semantics></math>$ by their canonical circle action. It seems that the only term in use with respect to this is “twisted loop space”, which however usually refers just to the constant loops $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ℒ</mi> <mi>const</mi></msub><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">(\mathcal{L}_{const}X)//S^1</annotation></semantics></math>$ . Since under nice conditions the derived functions on the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ℒ</mi><mi>Spec</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathcal{L}Spec(A)/S^1</annotation></semantics></math>$ is the cyclic homology complex of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>$ , I suggest that a good name is “cyclic loop space”. I made a quick note at cyclic loop space, just to fix and disambiguate terminology.
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeJun 21st 2021
- (edited Jun 21st 2021)
- PermaLink
Author: Urs
Format: MarkdownItexI have added pointer to cylic loop orbifolds in * [[Zhen Huan]], Def. 2.14 of: _Quasi-Elliptic Cohomology I_, Advances in Mathematics, Volume 337, 15 October 2018, Pages 107-138 ([arXiv:1805.06305](https://arxiv.org/abs/1805.06305), [doi:10.1016/j.aim.2018.08.007](https://doi.org/10.1016/j.aim.2018.08.007)) * [[Zhen Huan]], Section 2.1.2 of: _Quasi-elliptic cohomology_, 2017 ([hdl](http://hdl.handle.net/2142/97268)) * [[Zhen Huan]], Section 2 of: *Quasi-theories* ([arXiv:1809.06651](https://arxiv.org/abs/1809.06651)) Also added pointer to our * [[Vincent Braunack-Mayer]], [[Hisham Sati]], [[Urs Schreiber]]: Section 2.2 of _[[schreiber:Gauge enhancement of Super M-Branes|Gauge enhancement of Super M-Branes via rational parameterized stable homotopy theory]]_, Communications in Mathematical Physics, **371** 197 (2019) ([doi:10.1007/s00220-019-03441-4](https://doi.org/10.1007/s00220-019-03441-4), [arXiv:1806.01115](https://arxiv.org/abs/1806.01115)) and * [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 3 of: *[[schreiber:T-Duality from super Lie n-algebra cocycles for super p-branes|T-Duality from super Lie $n$-algebra cocycles for super $p$-branes]]*, [ATMP Volume 22 (2018) Number 5](http://www.intlpress.com/site/pub/pages/journals/items/atmp/content/vols/0022/0005/), [doi:10.4310/ATMP.2018.v22.n5.a3](http://dx.doi.org/10.4310/ATMP.2018.v22.n5.a3), [arXiv:1611.06536](https://arxiv.org/abs/1611.06536)) from which some of the entry material had been taken. <a href="https://ncatlab.org/nlab/revision/diff/cyclic+loop+space/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/cyclic+loop+space/5">v5</a>, <a href="https://ncatlab.org/nlab/show/cyclic+loop+space">current</a>
I have added pointer to cylic loop orbifolds in
- Zhen Huan, Def. 2.14 of: Quasi-Elliptic Cohomology I, Advances in Mathematics, Volume 337, 15 October 2018, Pages 107-138 (arXiv:1805.06305, doi:10.1016/j.aim.2018.08.007)
- Zhen Huan, Section 2.1.2 of: Quasi-elliptic cohomology, 2017 (hdl)
- Zhen Huan, Section 2 of: Quasi-theories (arXiv:1809.06651)
Also added pointer to our
- Vincent Braunack-Mayer, Hisham Sati, Urs Schreiber: Section 2.2 of Gauge enhancement of Super M-Branes via rational parameterized stable homotopy theory, Communications in Mathematical Physics, 371 197 (2019) (doi:10.1007/s00220-019-03441-4, arXiv:1806.01115)
and
- Domenico Fiorenza, Hisham Sati, Urs Schreiber, Section 3 of: T-Duality from super Lie $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>$ -algebra cocycles for super $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>$ -branes, ATMP Volume 22 (2018) Number 5, doi:10.4310/ATMP.2018.v22.n5.a3, arXiv:1611.06536)
from which some of the entry material had been taken.

diff, v5, current
- CommentRowNumber3.
- CommentAuthorUrs
- CommentTimeJul 2nd 2021
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to: * [[Gunnar Carlsson]], [[Ralph Cohen]], *The cyclic groups and the free loop space*, Commentarii Mathematici Helvetici **62** (1987) 423–449 ([doi:10.1007/BF02564455](https://doi.org/10.1007/BF02564455), [dml:140092](https://eudml.org/doc/140092)) <a href="https://ncatlab.org/nlab/revision/diff/cyclic+loop+space/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/cyclic+loop+space/11">v11</a>, <a href="https://ncatlab.org/nlab/show/cyclic+loop+space">current</a>
added pointer to:
- Gunnar Carlsson, Ralph Cohen, The cyclic groups and the free loop space, Commentarii Mathematici Helvetici 62 (1987) 423–449 (doi:10.1007/BF02564455, dml:140092)
diff, v11, current
- CommentRowNumber4.
- CommentAuthorUrs
- CommentTimeJul 2nd 2021
- PermaLink
Author: Urs
Format: MarkdownItexalso: * [[Nancy Hingston]], *An Equivariant Model for the Free Loop Space of $S^N$*, American Journal of Mathematics **114** 1 (1992) 139-155 ([doi:10.2307/2374740](https://doi.org/10.2307/2374740), [jstor:2374740](https://www.jstor.org/stable/2374740)) <a href="https://ncatlab.org/nlab/revision/diff/cyclic+loop+space/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/cyclic+loop+space/11">v11</a>, <a href="https://ncatlab.org/nlab/show/cyclic+loop+space">current</a>
also:
- Nancy Hingston, An Equivariant Model for the Free Loop Space of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>S</mi> <mi>N</mi></msup></mrow><annotation encoding="application/x-tex">S^N</annotation></semantics></math>$ , American Journal of Mathematics 114 1 (1992) 139-155 (doi:10.2307/2374740, jstor:2374740)
diff, v11, current
- CommentRowNumber5.
- CommentAuthorUrs
- CommentTimeJul 2nd 2021
- PermaLink
Author: Urs
Format: MarkdownItexalso: * [[Urs Frauenfelder]], *Dihedral homology and the moon*, J. Fixed Point Theory Appl. **14** (2013) 55–69 ([arXiv:1204.4549](https://arxiv.org/abs/1204.4549), [doi:10.1007/s11784-013-0146-z](https://doi.org/10.1007/s11784-013-0146-z)) <a href="https://ncatlab.org/nlab/revision/diff/cyclic+loop+space/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/cyclic+loop+space/11">v11</a>, <a href="https://ncatlab.org/nlab/show/cyclic+loop+space">current</a>
also:
- Urs Frauenfelder, Dihedral homology and the moon, J. Fixed Point Theory Appl. 14 (2013) 55–69 (arXiv:1204.4549, doi:10.1007/s11784-013-0146-z)
diff, v11, current
- CommentRowNumber6.
- CommentAuthorUrs
- CommentTimeJul 2nd 2021
- (edited Jul 2nd 2021)
- PermaLink
Author: Urs
Format: MarkdownItexadded ([here](https://ncatlab.org/nlab/show/cyclic+loop+space#StableSplitting)) statement of the stable splitting theorem by Carlsson & Cohen 1987: $$ \Omega^\infty \Sigma^\infty \big( Maps(S^1, \Sigma X) \sslash S^1 \big) \;\; \simeq \;\; \big( \Omega^\infty \Sigma^\infty B S^1 \big) \times \underset{ k \in \mathbb{N}_{\geq 1} }{\prod} \Omega^\infty \Sigma^\infty \big( (E (\mathbb{Z}/k))_+ \wedge_{\mathbb{Z}/k} X^{\wedge^n} \big) $$ <a href="https://ncatlab.org/nlab/revision/diff/cyclic+loop+space/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/cyclic+loop+space/12">v12</a>, <a href="https://ncatlab.org/nlab/show/cyclic+loop+space">current</a>

added (here) statement of the stable splitting theorem by Carlsson & Cohen 1987:
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>Ω</mi> <mn>∞</mn></msup><msup><mi>Σ</mi> <mn>∞</mn></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>Maps</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo><mo>⫽</mo><msup><mi>S</mi> <mn>1</mn></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="0.27778em"/><mspace width="0.27778em"/><mo>≃</mo><mspace width="0.27778em"/><mspace width="0.27778em"/><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>Ω</mi> <mn>∞</mn></msup><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>B</mi><msup><mi>S</mi> <mn>1</mn></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>×</mo><munder><mo lspace="0.16667em" rspace="0.16667em">∏</mo><mrow><mi>k</mi><mo>∈</mo><msub><mi>ℕ</mi> <mrow><mo>≥</mo><mn>1</mn></mrow></msub></mrow></munder><msup><mi>Ω</mi> <mn>∞</mn></msup><msup><mi>Σ</mi> <mn>∞</mn></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mi>k</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mo>+</mo></msub><msub><mo>∧</mo> <mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mi>k</mi></mrow></msub><msup><mi>X</mi> <mrow><msup><mo>∧</mo> <mi>n</mi></msup></mrow></msup><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> \Omega^\infty \Sigma^\infty \big( Maps(S^1, \Sigma X) \sslash S^1 \big) \;\; \simeq \;\; \big( \Omega^\infty \Sigma^\infty B S^1 \big) \times \underset{ k \in \mathbb{N}_{\geq 1} }{\prod} \Omega^\infty \Sigma^\infty \big( (E (\mathbb{Z}/k))_+ \wedge_{\mathbb{Z}/k} X^{\wedge^n} \big) </annotation></semantics></math>$
diff, v12, current
- CommentRowNumber7.
- CommentAuthorUrs
- CommentTimeJul 2nd 2021
- PermaLink
Author: Urs
Format: MarkdownItexalso pointer to * [[Ralph Cohen]], p. 194 of: *A Model for the Free Loop Space of a Suspension*, in: [[Haynes Miller]], [[Douglas Ravenel]] (eds.), *Algebraic Topology*, Lecture Notes in Mathematics **1286**, Springer 1985 ([doi:10.1007/BFb0078743](https://link.springer.com/chapter/10.1007/BFb0078743)) where this result was announced <a href="https://ncatlab.org/nlab/revision/diff/cyclic+loop+space/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/cyclic+loop+space/12">v12</a>, <a href="https://ncatlab.org/nlab/show/cyclic+loop+space">current</a>
also pointer to
- Ralph Cohen, p. 194 of: A Model for the Free Loop Space of a Suspension, in: Haynes Miller, Douglas Ravenel (eds.), Algebraic Topology, Lecture Notes in Mathematics 1286, Springer 1985 (doi:10.1007/BFb0078743)
where this result was announced

diff, v12, current
- CommentRowNumber8.
- CommentAuthorUrs
- CommentTimeAug 18th 2021
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to: * [[David Chataur]], *A bordism approach to string topology*, International Mathematics Research Notices **2005** 46, (2005) ([arXiv:math/0306080](https://arxiv.org/abs/math/0306080), [doi:10.1155/IMRN.2005.2829](https://doi.org/10.1155/IMRN.2005.2829)) for the terminology "string space" (Section 4.8.1) <a href="https://ncatlab.org/nlab/revision/diff/cyclic+loop+space/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/cyclic+loop+space/13">v13</a>, <a href="https://ncatlab.org/nlab/show/cyclic+loop+space">current</a>
added pointer to:
- David Chataur, A bordism approach to string topology, International Mathematics Research Notices 2005 46, (2005) (arXiv:math/0306080, doi:10.1155/IMRN.2005.2829)
for the terminology “string space” (Section 4.8.1)

diff, v13, current
- CommentRowNumber9.
- CommentAuthorUrs
- CommentTimeNov 28th 2022
- PermaLink
Author: Urs
Format: MarkdownItexI have adjusted the wording in [this](https://ncatlab.org/nlab/show/cyclic+loop+space#StapletonTerminology) partagraph in the Idea-section, which previouly was somewhat mixed up. <a href="https://ncatlab.org/nlab/revision/diff/cyclic+loop+space/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/cyclic+loop+space/14">v14</a>, <a href="https://ncatlab.org/nlab/show/cyclic+loop+space">current</a>

I have adjusted the wording in this partagraph in the Idea-section, which previouly was somewhat mixed up.

diff, v14, current

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nLab > Latest Changes: cyclic loop space