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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeSep 21st 2011
- (edited Nov 7th 2012)
- PermaLink
Author: Urs
Format: MarkdownItexcreated [[spin^c structure]] and [[twisted spin^c structure]]

created spin^c structure and twisted spin^c structure
- CommentRowNumber2.
- CommentAuthordomenico_fiorenza
- CommentTimeSep 21st 2011
- PermaLink
Author: domenico_fiorenza
Format: MarkdownItexcreated a draft for [[integral Stiefel-Whitney class]]. in the exaple it is said (but not explicitly) that spin^c structures can be seen as [[twisted spin structure]]s.

created a draft for integral Stiefel-Whitney class. in the exaple it is said (but not explicitly) that spin^c structures can be seen as twisted spin structures.
- CommentRowNumber3.
- CommentAuthorUrs
- CommentTimeSep 21st 2011
- (edited Sep 21st 2011)
- PermaLink
Author: Urs
Format: MarkdownItexThanks! The entry on integral SW classes should maybe mention the term [[Bockstein homomorphism]]. (I don't have time right now...)

Thanks!

The entry on integral SW classes should maybe mention the term Bockstein homomorphism.

(I don’t have time right now…)
- CommentRowNumber4.
- CommentAuthordomenico_fiorenza
- CommentTimeSep 21st 2011
- (edited Sep 21st 2011)
- PermaLink
Author: domenico_fiorenza
Format: MarkdownItexyes it should, but we had not [[Bockstein homomorphism]] and I had no time for creating it then. But I have now, so.. edit: now we have [[Bockstein homomorphism]]

yes it should, but we had not Bockstein homomorphism and I had no time for creating it then. But I have now, so..

edit: now we have Bockstein homomorphism
- CommentRowNumber5.
- CommentAuthorUrs
- CommentTimeSep 21st 2011
- PermaLink
Author: Urs
Format: MarkdownItexGreat, thanks! I have added a few more hyperlinks and section titles, etc. I have also added an Examples-section at [[Bockstein homomorphism]], mentioning also the relation to Steenrod squares. You said you regard a $spin^c$-structure as an example of a twisted $spin$-structure. I see this at a heuristic level. Is it also true in the formal sense? Maybe I am being dense here.

Great, thanks!

I have added a few more hyperlinks and section titles, etc. I have also added an Examples-section at Bockstein homomorphism, mentioning also the relation to Steenrod squares.

You said you regard a $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">spin^c</annotation></semantics></math>$ -structure as an example of a twisted $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>spin</mi></mrow><annotation encoding="application/x-tex">spin</annotation></semantics></math>$ -structure. I see this at a heuristic level. Is it also true in the formal sense? Maybe I am being dense here.
- CommentRowNumber6.
- CommentAuthordomenico_fiorenza
- CommentTimeSep 22nd 2011
- PermaLink
Author: domenico_fiorenza
Format: MarkdownItexI've now expanded [[Bockstein homomorphism]]. >You said you regard a $spin^c$-structure as an example of a twisted $spin$-structure. I see this at a heuristic level. Is it also true in the formal sense? yes (I think). the connecting morphism $\mathbf{B}^2\mathbb{Z}_2\to \mathbf{B}^3\mathbb{Z}$ is naturally identified with the natural morphism $\mathbf{B}^2\mathbb{Z}_2\to \mathbf{B}^2 U(1)$ induced by the inclusion of the subgroup $\{\pm1\}$ in $U(1)$ (I'm now writing this at [[Bockstein homomorphism]]). So we have that a $spin^c$-structure on $X$ is a trivialization of $W_3\circ T X: X \to \mathbf{B}^3\mathbb{Z}\simeq \mathbf{B}^2 U(1)$; by definition of $W_3$, this is a trivialization of $\beta\circ w_2\circ T X: X \to \mathbf{B}^2 U(1)$, and this is in turn equivalent to a factorization of $w_2\circ T X: X \to \mathbf{B}^2 \mathbb{Z}_2$ through the [[homotopy fiber]] of $\beta:\mathbf{B}^2\mathbb{Z}_2\to \mathbf{B}^2 U(1)$, which is $\mathbf{B} U(1)$ by the [[fiber sequence]] $\cdots \to \mathbf{B}^n \mathbb{Z}_2\to \mathbf{B}^n U(1)\to \mathbf{B}^n U(1) \to \mathbf{B}^{n+1} \mathbb{Z}_2\to \cdots$. So $spin^c$-structures on $X$ are identified with lifts of $w_2\circ T X: X \to \mathbf{B}^2 \mathbb{Z}_2$ to $\mathbf{B}U(1)$. These are in turn equivalent to homotopy commutative diagrams $$ \begin{aligned} X& \to & \mathbf{B} U(1)\\ \downarrow_{T X} & & \downarrow_{c_1 \mod 2}\\ \mathbf{B}SO&\stackrel{w_2}{\to} & \mathbf{B}^2 \mathbb{Z}_2 \end{aligned} $$ Hence $\mathbf{B} Spin^c$ is the homotopy pullback $$ \begin{aligned} \mathbf{B} Spin^c& \to & \mathbf{B} U(1)\\ \downarrow & & \downarrow_{c_1 \mod 2}\\ \mathbf{B}SO&\stackrel{w_2}{\to} & \mathbf{B}^2 \mathbb{Z}_2 \end{aligned} $$ whereas $\mathbf{B} Spin$ is the homotopy fiber $$ \begin{aligned} \mathbf{B} Spin& \to & *\\ \downarrow & & \downarrow\\ \mathbf{B}SO&\stackrel{w_2}{\to} & \mathbf{B}^2 \mathbb{Z}_2 \end{aligned} $$

I’ve now expanded Bockstein homomorphism.

You said you regard a $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">spin^c</annotation></semantics></math>$ -structure as an example of a twisted $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>spin</mi></mrow><annotation encoding="application/x-tex">spin</annotation></semantics></math>$ -structure. I see this at a heuristic level. Is it also true in the formal sense?

yes (I think). the connecting morphism $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}^2\mathbb{Z}_2\to \mathbf{B}^3\mathbb{Z}</annotation></semantics></math>$ is naturally identified with the natural morphism $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^2\mathbb{Z}_2\to \mathbf{B}^2 U(1)</annotation></semantics></math>$ induced by the inclusion of the subgroup $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">{</mo><mo>±</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\pm1\}</annotation></semantics></math>$ in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>$ (I’m now writing this at Bockstein homomorphism). So we have that a $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">spin^c</annotation></semantics></math>$ -structure on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>$ is a trivialization of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi> <mn>3</mn></msub><mo>∘</mo><mi>T</mi><mi>X</mi><mo>:</mo><mi>X</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>ℤ</mi><mo>≃</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W_3\circ T X: X \to \mathbf{B}^3\mathbb{Z}\simeq \mathbf{B}^2 U(1)</annotation></semantics></math>$ ; by definition of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">W_3</annotation></semantics></math>$ , this is a trivialization of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>∘</mo><msub><mi>w</mi> <mn>2</mn></msub><mo>∘</mo><mi>T</mi><mi>X</mi><mo>:</mo><mi>X</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\beta\circ w_2\circ T X: X \to \mathbf{B}^2 U(1)</annotation></semantics></math>$ , and this is in turn equivalent to a factorization of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>w</mi> <mn>2</mn></msub><mo>∘</mo><mi>T</mi><mi>X</mi><mo>:</mo><mi>X</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">w_2\circ T X: X \to \mathbf{B}^2 \mathbb{Z}_2</annotation></semantics></math>$ through the homotopy fiber of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>:</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\beta:\mathbf{B}^2\mathbb{Z}_2\to \mathbf{B}^2 U(1)</annotation></semantics></math>$ , which is $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B} U(1)</annotation></semantics></math>$ by the fiber sequence $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>⋯</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>→</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex">\cdots \to \mathbf{B}^n \mathbb{Z}_2\to \mathbf{B}^n U(1)\to \mathbf{B}^n U(1) \to \mathbf{B}^{n+1} \mathbb{Z}_2\to \cdots</annotation></semantics></math>$ . So $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">spin^c</annotation></semantics></math>$ -structures on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>$ are identified with lifts of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>w</mi> <mn>2</mn></msub><mo>∘</mo><mi>T</mi><mi>X</mi><mo>:</mo><mi>X</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">w_2\circ T X: X \to \mathbf{B}^2 \mathbb{Z}_2</annotation></semantics></math>$ to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}U(1)</annotation></semantics></math>$ . These are in turn equivalent to homotopy commutative diagrams
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msub><mo stretchy="false">↓</mo> <mrow><mi>T</mi><mi>X</mi></mrow></msub></mtd> <mtd/> <mtd><msub><mo stretchy="false">↓</mo> <mrow><msub><mi>c</mi> <mn>1</mn></msub><mo lspace="0.22222em" rspace="0.22222em">mod</mo><mn>2</mn></mrow></msub></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>w</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} X&amp; \to &amp; \mathbf{B} U(1)\\ \downarrow_{T X} &amp; &amp; \downarrow_{c_1 \mod 2}\\ \mathbf{B}SO&amp;\stackrel{w_2}{\to} &amp; \mathbf{B}^2 \mathbb{Z}_2 \end{aligned} </annotation></semantics></math>$
Hence $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{B} Spin^c</annotation></semantics></math>$ is the homotopy pullback
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd/> <mtd><msub><mo stretchy="false">↓</mo> <mrow><msub><mi>c</mi> <mn>1</mn></msub><mo lspace="0.22222em" rspace="0.22222em">mod</mo><mn>2</mn></mrow></msub></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>w</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathbf{B} Spin^c&amp; \to &amp; \mathbf{B} U(1)\\ \downarrow &amp; &amp; \downarrow_{c_1 \mod 2}\\ \mathbf{B}SO&amp;\stackrel{w_2}{\to} &amp; \mathbf{B}^2 \mathbb{Z}_2 \end{aligned} </annotation></semantics></math>$
whereas $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} Spin</annotation></semantics></math>$ is the homotopy fiber
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd/> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>w</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathbf{B} Spin&amp; \to &amp; *\\ \downarrow &amp; &amp; \downarrow\\ \mathbf{B}SO&amp;\stackrel{w_2}{\to} &amp; \mathbf{B}^2 \mathbb{Z}_2 \end{aligned} </annotation></semantics></math>$
- CommentRowNumber7.
- CommentAuthorUrs
- CommentTimeSep 22nd 2011
- PermaLink
Author: Urs
Format: MarkdownItexHi Domenico, thanks! Good point. You observe that $Spin^c$ is the homotopy pullback of $w_2$ along $c_1 mod 2$. I have now writte out a very detailed proof of this [here](http://ncatlab.org/nlab/show/spin%5Ec+structure#PropertiesOfSpinC). Please check. But I have one slight disagreement: maybe we should generalize our definition, but with what we used to say (for instance at [[twisted differential c-structure]]) it seems not quite right to say that a $Spin^c$-structure is a twisted $Spin$-structure. Unless I am missing somethig. Because a twisted $Spin$-structure is defined to be something in the homotopy pullback $$ \array{ w_2 Struc_{tw}(X) &\to& H^2(X, \mathbb{Z}_2) \\ \downarrow && \downarrow \\ \mathbf{H}(X, \mathbf{B} SO) &\stackrel{w_2}{\to}& \mathbf{H}(X, \mathbf{B}^2 \mathbb{Z}_2) } \,. $$ Here we are pulling back a 0-truncated object in the top right. For fitting your observation into a notion of twisted Spin-structures one would have to generalize that definition. Maybe one should! But as long as we haven't done so "officially", let's maybe be careful with the terminology. But let me know if I am missing some point.

Hi Domenico,

thanks! Good point. You observe that $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>$ is the homotopy pullback of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>w</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">w_2</annotation></semantics></math>$ along $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mi>mod</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">c_1 mod 2</annotation></semantics></math>$ .

I have now writte out a very detailed proof of this here. Please check.

But I have one slight disagreement: maybe we should generalize our definition, but with what we used to say (for instance at twisted differential c-structure) it seems not quite right to say that a $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>$ -structure is a twisted $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Spin</mi></mrow><annotation encoding="application/x-tex">Spin</annotation></semantics></math>$ -structure. Unless I am missing somethig. Because a twisted $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Spin</mi></mrow><annotation encoding="application/x-tex">Spin</annotation></semantics></math>$ -structure is defined to be something in the homotopy pullback
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>w</mi> <mn>2</mn></msub><msub><mi>Struc</mi> <mi>tw</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>H</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd/> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>w</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="0.16667em"/><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ w_2 Struc_{tw}(X) &amp;\to&amp; H^2(X, \mathbb{Z}_2) \\ \downarrow &amp;&amp; \downarrow \\ \mathbf{H}(X, \mathbf{B} SO) &amp;\stackrel{w_2}{\to}&amp; \mathbf{H}(X, \mathbf{B}^2 \mathbb{Z}_2) } \,. </annotation></semantics></math>$
Here we are pulling back a 0-truncated object in the top right. For fitting your observation into a notion of twisted Spin-structures one would have to generalize that definition. Maybe one should! But as long as we haven’t done so “officially”, let’s maybe be careful with the terminology.

But let me know if I am missing some point.
- CommentRowNumber8.
- CommentAuthordomenico_fiorenza
- CommentTimeSep 22nd 2011
- PermaLink
Author: domenico_fiorenza
Format: MarkdownItexHi Urs, >I have now writte out a very detailed proof of this here. Please check. Looks fine. > But I have one slight disagreement: maybe we should generalize our definition, but with what we used to say (for instance at twisted differential c-structure) it seems not quite right to say that a $Spin^c$-structure is a twisted $Spin$-structure. Right. You know I'd prefer having a more flexible notion of twisted cohomology, using an arbitrary morphism $c:C\to \mathbf{H}(X,A)$ to twist. So in teh case of $Spin^c$, the twisting morphisms would be $c_1 \mod 2: \mathbf{H}(X,\mathbf{B} U(1)) \to \mathbf{H}(X,\mathbf{B}2 \mathbb{Z})$, and so $Spin^c$-structure would be "$(c_1 \mod 2)$-twisted $Spin$-structures". But I agree we should reserve the "absolute" name "twisted cohomlogy" for the twisting morphism $H(X,A)\to \mathbf{H}(X,A)$, so you're right.

Hi Urs,

I have now writte out a very detailed proof of this here. Please check.

Looks fine.

But I have one slight disagreement: maybe we should generalize our definition, but with what we used to say (for instance at twisted differential c-structure) it seems not quite right to say that a $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>$ -structure is a twisted $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Spin</mi></mrow><annotation encoding="application/x-tex">Spin</annotation></semantics></math>$ -structure.

Right. You know I’d prefer having a more flexible notion of twisted cohomology, using an arbitrary morphism $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>:</mo><mi>C</mi><mo>→</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c:C\to \mathbf{H}(X,A)</annotation></semantics></math>$ to twist. So in teh case of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>$ , the twisting morphisms would be $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo lspace="0.22222em" rspace="0.22222em">mod</mo><mn>2</mn><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mn>2</mn><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c_1 \mod 2: \mathbf{H}(X,\mathbf{B} U(1)) \to \mathbf{H}(X,\mathbf{B}2 \mathbb{Z})</annotation></semantics></math>$ , and so $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>$ -structure would be “ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo lspace="0.22222em" rspace="0.22222em">mod</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(c_1 \mod 2)</annotation></semantics></math>$ -twisted $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Spin</mi></mrow><annotation encoding="application/x-tex">Spin</annotation></semantics></math>$ -structures”. But I agree we should reserve the “absolute” name “twisted cohomlogy” for the twisting morphism $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H(X,A)\to \mathbf{H}(X,A)</annotation></semantics></math>$ , so you’re right.
- CommentRowNumber9.
- CommentAuthorUrs
- CommentTimeSep 22nd 2011
- PermaLink
Author: Urs
Format: MarkdownItex> Looks fine. Thanks. I'll further expand on the second bit in a little while. Am awefully busy today with other things.

Looks fine.

Thanks. I’ll further expand on the second bit in a little while. Am awefully busy today with other things.
- CommentRowNumber10.
- CommentAuthorUrs
- CommentTimeSep 22nd 2011
- (edited Sep 22nd 2011)
- PermaLink
Author: Urs
Format: MarkdownItex> You know I'd prefer having a more flexible notion of twisted cohomology, Yes, I know. And there is certainly a good point to be made. For instance the Hopkins-Singer definition of differential cohomology for unstable coefficients is a definition of twisted cohomology in the "restrictive" sense. But they have this parameter "$s$" in their definitions. This is really the truncation degree for the thing being pulled back. For $s \geq 1$ this is an example of the "more general notion of differential cohomology".

You know I’d prefer having a more flexible notion of twisted cohomology,

Yes, I know. And there is certainly a good point to be made.

For instance the Hopkins-Singer definition of differential cohomology for unstable coefficients is a definition of twisted cohomology in the “restrictive” sense. But they have this parameter “ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math>$ ” in their definitions. This is really the truncation degree for the thing being pulled back. For $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">s \geq 1</annotation></semantics></math>$ this is an example of the “more general notion of differential cohomology”.
- CommentRowNumber11.
- CommentAuthorUrs
- CommentTimeSep 22nd 2011
- PermaLink
Author: Urs
Format: MarkdownItexOkay, I am through with typing up what I think is the fully detailed proof that $$ \mathbf{B}Spin^c \to \mathbf{B}SO \stackrel{\mathbf{W}_3}{\to} \mathbf{B}^2 U(2) $$ is a [[fiber sequence]] in $Smooth \infty Grpd$, following your (Domenico's) indications above. See at [[spin^c]] the new section [As homotopy fiber of smooth W3](AsHomotopyFiberOfSmoothW3). But check. I am a bit in a haste. Not really the right context to write out proofs. Check carefully. I try to come to this later this evening and polish if necessary.

Okay, I am through with typing up what I think is the fully detailed proof that
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>W</mi></mstyle> <mn>3</mn></msub></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}Spin^c \to \mathbf{B}SO \stackrel{\mathbf{W}_3}{\to} \mathbf{B}^2 U(2) </annotation></semantics></math>$
is a fiber sequence in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">Smooth \infty Grpd</annotation></semantics></math>$ , following your (Domenico’s) indications above.

See at spin^c the new section As homotopy fiber of smooth W3.

But check. I am a bit in a haste. Not really the right context to write out proofs. Check carefully. I try to come to this later this evening and polish if necessary.
- CommentRowNumber12.
- CommentAuthorUrs
- CommentTimeNov 7th 2012
- PermaLink
Author: Urs
Format: MarkdownItexI have started adding a little bit to _[[spin^c structure]]_; two more references and a paragraph on inducing $Spin^c$-structures from almost complex structures.

I have started adding a little bit to spin^c structure; two more references and a paragraph on inducing $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>$ -structures from almost complex structures.
- CommentRowNumber13.
- CommentAuthorUrs
- CommentTimeNov 9th 2012
- PermaLink
Author: Urs
Format: MarkdownItexFilled in the missing details at _[Spin^c structure -- From almost complex structure](http://ncatlab.org/nlab/show/spin%5Ec+structure#FromAlmostComplexStructures)_.

Filled in the missing details at Spin^c structure – From almost complex structure.
- CommentRowNumber14.
- CommentAuthornLab edit announcer
- CommentTimeMay 8th 2019
- PermaLink
Author: nLab edit announcer
Format: MarkdownItexattempting to make "spin^c" in internal links format properly (i.e. with a superscript), but I don't know if this will work. If not, I'll revert. Arun Debray <a href="https://ncatlab.org/nlab/revision/diff/spin%5Ec+structure/23">diff</a>, <a href="https://ncatlab.org/nlab/revision/spin%5Ec+structure/23">v23</a>, <a href="https://ncatlab.org/nlab/show/spin%5Ec+structure">current</a>

attempting to make “spin^c” in internal links format properly (i.e. with a superscript), but I don’t know if this will work. If not, I’ll revert.

Arun Debray

diff, v23, current
- CommentRowNumber15.
- CommentAuthornLab edit announcer
- CommentTimeMay 8th 2019
- PermaLink
Author: nLab edit announcer
Format: MarkdownItexIt looks like one can write `spinc` inside the link text, so I did that for the occurrences of spin^c in internal links in this document. Arun Debray <a href="https://ncatlab.org/nlab/revision/diff/spin%5Ec+structure/23">diff</a>, <a href="https://ncatlab.org/nlab/revision/spin%5Ec+structure/23">v23</a>, <a href="https://ncatlab.org/nlab/show/spin%5Ec+structure">current</a>

It looks like one can write spinc inside the link text, so I did that for the occurrences of spin^c in internal links in this document.

Arun Debray

diff, v23, current
- CommentRowNumber16.
- CommentAuthorUrs
- CommentTimeMay 8th 2019
- PermaLink
Author: Urs
Format: MarkdownItexThanks! Good point. Will try to stick to that, too.

Thanks! Good point. Will try to stick to that, too.
- CommentRowNumber17.
- CommentAuthorUrs
- CommentTimeJul 5th 2024
- PermaLink
Author: Urs
Format: MarkdownItexchanged page name (from "`spin^c structure`" to "`spinᶜ structure`") for better looks, following discussion [here](https://nforum.ncatlab.org/discussion/18215/title-convention/?Focus=117893#Comment_117893) <a href="https://ncatlab.org/nlab/revision/diff/spin%E1%B6%9C+structure/28">diff</a>, <a href="https://ncatlab.org/nlab/revision/spin%E1%B6%9C+structure/28">v28</a>, <a href="https://ncatlab.org/nlab/show/spin%E1%B6%9C+structure">current</a>

changed page name (from “spin^c structure” to “spinᶜ structure”) for better looks, following discussion here

diff, v28, current
- CommentRowNumber18.
- CommentAuthorSamuel Adrian Antz
- CommentTimeJul 5th 2024
- PermaLink
Author: Samuel Adrian Antz
Format: MarkdownItexChanged “spin^c” to “spinᶜ” and “spin^h” to “spinʰ” in related concepts. (Links will be checked.) It was already displayed correctly for some links with a case split, so now the code is shortened. <a href="https://ncatlab.org/nlab/revision/diff/spin%E1%B6%9C+structure/29">diff</a>, <a href="https://ncatlab.org/nlab/revision/spin%E1%B6%9C+structure/29">v29</a>, <a href="https://ncatlab.org/nlab/show/spin%E1%B6%9C+structure">current</a>

Changed “spin^c” to “spinᶜ” and “spin^h” to “spinʰ” in related concepts. (Links will be checked.) It was already displayed correctly for some links with a case split, so now the code is shortened.

diff, v29, current

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