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created spin^c structure and twisted spin^c structure
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.
Thanks!
The entry on integral SW classes should maybe mention the term Bockstein homomorphism.
(I don’t have time right now…)
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
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 $spin^c$-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.
I’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
$\begin{aligned} \mathbf{B} Spin& \to & *\\ \downarrow & & \downarrow\\ \mathbf{B}SO&\stackrel{w_2}{\to} & \mathbf{B}^2 \mathbb{Z}_2 \end{aligned}$Hi 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. 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 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.
Looks fine.
Thanks. I’ll further expand on the second bit in a little while. Am awefully busy today with other things.
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”.
Okay, 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.
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.
I 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.
Filled in the missing details at Spin^c structure – From almost complex structure.
Thanks! Good point. Will try to stick to that, too.
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