Obstruction theoretic approach to complex spin structures

It seems this question is extremely well-known in literature, but I have poor understanding on these stuffs, and it seems nobody asks this question before in math.stackexchange, so I decided to ask this:

It is usually said that the 2nd Stiefel-Whitney class is the obstruction to the existence of spin structures on a vector bundle, and if it vanishes, then the set of spin structures becomes an affine space over the 1st cohomology group of the base space. One explanation uses the serre LES

$0\to H^1(X;\mathbb{Z}_2)\to H^1(P_{SO(n)};\mathbb{Z}_2)\to H^1(SO(n);\mathbb{Z}_2)\to H^2(X;\mathbb{Z}_2)$

where the image of the unique nontrivial element $a\in H^1(SO(n);\mathbb{Z}_2)\cong \mathbb{Z}_2)$ by the last map is the 2nd Stiefel-Whitney class of the bundle. The virtue of this picture is the LES used above explains both the obstruction to and the classification(i.e., the affine structure) of spin structures.

Okay, this is great, and now move on to the complex spin case. We know that the integral 3rd Stiefel-Whitney class $W_3=\beta_2w_2$, where $w_2$ is the 2nd Stiefel-Whitney class and $\beta_2:H^2(X;\mathbb{Z}_2)\to H^3(X;\mathbb{Z})$ is the relevant Bockstein homomorphism., is the obstruction to the existence of complex spin structures on a vector bundle over $X$. And we also know that if this vanishes, then the set of all complex spin structures is classified as an $H^2(X;\mathbb{Z})$-affine space.

One proof that I know of is rather ad-hoc. First, we can define the map

(the set of $spin^c$ structures)$\to \left\{x\in H^2(Xl\mathbb{Z}):x\equiv w_2\mathrm{\;mod\;}2\right\}$

which is taking the 1st chern class of the determinant bundle of the $spin^c$ structure. It is easy to check that this map is surjective considering one of the equivalent definition of $spin^c$ structures that a $spin^c$ structure on a vector bundle $E$ is equivalent to a choice of complex line bundle $L$ and a spin structure on $E\oplus L$(up to gauge transformation on $L$), and the above LES. And then, for a fixed $x$, the action of gauge transformation on $L$ identifies two spin structures on $E\oplus L$ which differ by the kernel of $\beta_1:H^1(X;\mathbb{Z}_2)\to H^2(X;\mathbb{Z})$, so over each $x$ there are $H^1(X;\mathbb{Z}_2)/\mathrm{ker}\beta_1=\mathrm{im}\beta_1$-worth many $spin^c$ structures, and clearly the set on the RHS is $H^2(X;\mathbb{Z})/\mathrm{im}\beta_1$, giving the desired statement.

What I want to know is an obstruction theoretic alternative to this picture. It is often stated without details in many literature, especially the part of obtaining affine structure on the $spin^c$ structures seems quite mysterious to me. (I have literally no idea how the proof goes.) I especially am interested in the case of obstructions to extending $spin^c$ structures given on the boundary to the interior, which seems is suitable to use the tool of obstruction theory, but any suggestion or reference would be appreciated.

My main interest lies on understanding the relative obstruction to extending a given $spin^c$ structure on the boundary $\partial M$ of a differentiable manifold $M$ to the interior, which can be summarized as

Proposition. There exists a $spin^c$ structure extending a $spin^c$ structure $\mathfrak{s}$ on the boundary (which can be identified with a $spin^c$ structure on $TM|_{\partial M}$) if and only if a certain cohomology class $W(M,\mathfrak{s})\in H^3(M,\partial M)$ vanishes. In that case, the set of $spin^c$ structures on $M$ extending $\mathfrak{s}$ is a $H^2(M,\partial)$-affine space.

And I found out hard to modify the arguments that I explained above to prove this proposition, but it is naturally expected that an obstruction theoretic method can be easily modified (or I hope so?). Therefore either an obstruction theoretic alternatives of the above or a modification of the above to the relative case are welcomed.

1 Answer

I'm not sure this is exactly what you're looking for, and neither is it complete, but as I've finished typing it and its too long for a comment, I'll post it. I will refer you to Gompf's paper $Spin^c$-structures and homotopy equivalences at https://arxiv.org/pdf/math/9705218.pdf for a better discussion (especially his proposition 1)

Use the fibration sequence $S^1\xrightarrow{j} Spin^c_n\xrightarrow{p} SO_n$ where $p:Spin^c_n=Spin_n\times_{\mathbb{Z}_2}S^1\rightarrow SO(n)$ is induced by the $\mathbb{Z}_2$-invariant composition $Spin_n\times S^1\xrightarrow{pr}Spin\xrightarrow{\pi}SO_n$ with $pi$ the universal cover.

Now proceed from there: classify the fibration and deloop to get a homotopy fibration

$BSpin_n^c\xrightarrow{Bp}BSO_n\xrightarrow{\beta_2\omega_2} K(\mathbb{Z},3)$

Now we have a map $f:M\rightarrow BSO_n$ classifying our given bundle (which we assume to be orientable since we are asking for a $Spin^c_n$-structure) and the composition $(\beta_2\omega_2)\circ f=f^*(\beta_2\omega_2)=\beta_2f^*(\omega_2):M\rightarrow K(\mathbb{Z},3)$ becomes null-homotopic when restricted to $i:\partial M\hookrightarrow M$. This is because $f\circ i:\partial M\rightarrow BSO_n$ lifts through $Bp$ to a map $\tilde{f}:\partial M\rightarrow BSpin^c_n$ defining the $Spin_n^c$-structure on $TM|_{\partial M}$.

It follows that there exists some map $\theta:C_i\rightarrow K(\mathbb{Z},3)$ from the (homotopy) cofiber of the inclusion $i:\partial M\hookrightarrow M$ such that $\theta\circ q=q^*\theta=\beta_2f^*\omega_2$, where $q:M\rightarrow C_i$ is the projection. In this way we quantify the obstruction to extending the $Spin^c_n$-structure on $\partial M$ by the class

$\theta\in H^3(C_i;\mathbb{Z})\cong H^3(M,\partial M)$.

In your notation this is the class $W(M,\mathfrak{s})$. If this class vanishes then there exists some map $\tilde{f}':M\rightarrow BSpin_n^c$ lifting $f$ and extending $\tilde{f}$, which thus defines the $Spin_n^c$-structure you seek.

Now if $\tilde{f}'_1,\tilde{f}'_2:M\rightarrow BSpin_n^c$ are two maps as above then they differ by some map $\eta:M\rightarrow BS^1$ such that $\tilde{f}'_1=\tilde{f}'_2+Bj\circ \eta$ (recall $j$ from the opening paragraph). Here I have used the principal action of $BS^1$ on $BSpin_n^c$ induced by fibration, and denoted it by addition.

Now sending the pair $\tilde{f}'_1$, $\tilde{f}'_2$ to the class $\eta\in H^2(M)$ defines the basic affine structure. I haven't shown that this map is well-defined, since it depends upon the choice of $\eta$. I'm not sure the proposition you give is true without some simple assumptions on the manifold, but I'm guessing that with a little bit more information it can be shown that $\eta$ comes from a unique class in $H^2(M,\partial M)$ which is independent of any choices.

• Your answer is really helpful, thanks!! Actually I noticed the paper by Gompf you mentioned here but it was rather elusive before seeing your kind elaboration. BTW, I think $\eta$ already belongs to $H^2(M,\partial M)$ for the two pair coincides on $\partial M$ which makes $\eta|_{\partial M}:\partial M\to BS^1$ nullhomotopic, isn't it? Aug 23 '17 at 20:20
• I only managed to show that the composite $Bj\circ\eta\circ i$ was null-homotopic, so $i^*\eta$ could potentially be non-trivial. Requiring it to be trivial you could get a class in $H^2(M,\partial M)$, but I couldn't show its uniqueness without placing some assumptions on the manifold. Aug 24 '17 at 7:34
• Hmm, I see. It seems the hullhomotopic-at-boundary condition on $\eta$ makes some trouble to obtain an $H^2(M)$-action. The proposition seems is true without further assumption on the "niceness" of $M$, like insisted in here, proposition 1.2. But of course they give no proof :( But still you showed me what is going on under the water, thanks again. Aug 24 '17 at 7:54
• My apology for disturbing you, I think in the paper that I hyperlinked in the comment their main interest is on 3-manifolds, hence $TM|_\partial M$ is trivial and the classifying map restricted on the boundary should be nullhomotopic. And the same holds for 4-manifolds, explaining why the relative complex spins have affine action by $H^2(M,\partial M)$ in many literature in low-dimensional topology. Now I starts to doubt that the proposition without such a dimension assumption would be generally true. Aug 24 '17 at 8:25