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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 21st 2011
    • (edited Nov 7th 2012)
  1. 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)

    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)

    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

    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 cspin^c-structure as an example of a twisted spinspin-structure. I see this at a heuristic level. Is it also true in the formal sense? Maybe I am being dense here.

  2. I’ve now expanded Bockstein homomorphism.

    You said you regard a spin cspin^c-structure as an example of a twisted spinspin-structure. I see this at a heuristic level. Is it also true in the formal sense?

    yes (I think). the connecting morphism B 2 2B 3\mathbf{B}^2\mathbb{Z}_2\to \mathbf{B}^3\mathbb{Z} is naturally identified with the natural morphism B 2 2B 2U(1)\mathbf{B}^2\mathbb{Z}_2\to \mathbf{B}^2 U(1) induced by the inclusion of the subgroup {±1}\{\pm1\} in U(1)U(1) (I’m now writing this at Bockstein homomorphism). So we have that a spin cspin^c-structure on XX is a trivialization of W 3TX:XB 3B 2U(1)W_3\circ T X: X \to \mathbf{B}^3\mathbb{Z}\simeq \mathbf{B}^2 U(1); by definition of W 3W_3, this is a trivialization of βw 2TX:XB 2U(1)\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 2TX:XB 2 2w_2\circ T X: X \to \mathbf{B}^2 \mathbb{Z}_2 through the homotopy fiber of β:B 2 2B 2U(1)\beta:\mathbf{B}^2\mathbb{Z}_2\to \mathbf{B}^2 U(1), which is BU(1)\mathbf{B} U(1) by the fiber sequence B n 2B nU(1)B nU(1)B n+1 2\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 cspin^c-structures on XX are identified with lifts of w 2TX:XB 2 2w_2\circ T X: X \to \mathbf{B}^2 \mathbb{Z}_2 to BU(1)\mathbf{B}U(1). These are in turn equivalent to homotopy commutative diagrams

    X BU(1) TX c 1mod2 BSO w 2 B 2 2 \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 BSpin c\mathbf{B} Spin^c is the homotopy pullback

    BSpin c BU(1) c 1mod2 BSO w 2 B 2 2 \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 BSpin\mathbf{B} Spin is the homotopy fiber

    BSpin * BSO w 2 B 2 2 \begin{aligned} \mathbf{B} Spin& \to & *\\ \downarrow & & \downarrow\\ \mathbf{B}SO&\stackrel{w_2}{\to} & \mathbf{B}^2 \mathbb{Z}_2 \end{aligned}
    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 22nd 2011

    Hi Domenico,

    thanks! Good point. You observe that Spin cSpin^c is the homotopy pullback of w 2w_2 along c 1mod2c_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 cSpin^c-structure is a twisted SpinSpin-structure. Unless I am missing somethig. Because a twisted SpinSpin-structure is defined to be something in the homotopy pullback

    w 2Struc tw(X) H 2(X, 2) H(X,BSO) w 2 H(X,B 2 2). \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.

  3. 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 cSpin^c-structure is a twisted SpinSpin-structure.

    Right. You know I’d prefer having a more flexible notion of twisted cohomology, using an arbitrary morphism c:CH(X,A)c:C\to \mathbf{H}(X,A) to twist. So in teh case of Spin cSpin^c, the twisting morphisms would be c 1mod2:H(X,BU(1))H(X,B2)c_1 \mod 2: \mathbf{H}(X,\mathbf{B} U(1)) \to \mathbf{H}(X,\mathbf{B}2 \mathbb{Z}), and so Spin cSpin^c-structure would be “(c 1mod2)(c_1 \mod 2)-twisted SpinSpin-structures”. But I agree we should reserve the “absolute” name “twisted cohomlogy” for the twisting morphism H(X,A)H(X,A)H(X,A)\to \mathbf{H}(X,A), so you’re right.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 22nd 2011

    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)

    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 “ss” in their definitions. This is really the truncation degree for the thing being pulled back. For s1s \geq 1 this is an example of the “more general notion of differential cohomology”.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 22nd 2011

    Okay, I am through with typing up what I think is the fully detailed proof that

    BSpin cBSOW 3B 2U(2) \mathbf{B}Spin^c \to \mathbf{B}SO \stackrel{\mathbf{W}_3}{\to} \mathbf{B}^2 U(2)

    is a fiber sequence in SmoothGrpdSmooth \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.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeNov 7th 2012

    I have started adding a little bit to spin^c structure; two more references and a paragraph on inducing Spin cSpin^c-structures from almost complex structures.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeNov 9th 2012

    Filled in the missing details at Spin^c structure – From almost complex structure.