Recently I asked on stack exchange the following question: https://math.stackexchange.com/questions/2088888/vanishing-of-certain-cohomology-class-and-existence-of-spin-structure

I would like to know whether there is a similar construction for the spin$^c$ manifolds: namely whether one can construct some cohomology class indpendent from the choice of transition functions and liftings to $spin^c(n)$ with the property that this class is trivial iff $M$ is spin$^c$ manifold.

Remark: I know $C^*$-algebraic approach which gives the so called Dixmier-Douady class in $H^3(M,\mathbb{Z})$. However I would like to understand whether one can proceed purely geometrically.


1 Answer 1


The obstruction to spin$^c$ is often denoted by $W_3(M) \in H^3(M,\mathbb{Z})$. It is obtained as follows: Let $$ \beta \colon H^2(M,\mathbb{Z}/2\mathbb{Z}) \to H^3(M,\mathbb{Z}) $$ be the Bockstein homomorphism obtained from the short exact sequence $$ 0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0 $$ Then we have $W_3(M) = \beta(w_2(M))$, i.e. the class obtained by applying the Bockstein homomorphism to the second Stiefel-Whitney class. This agrees with the Dixmier-Douady class of the (stabilisation of the) complex Clifford bundle $\mathbb{C}l(M)$.

Geometrically you can do the following: There is an exact sequence $$ 1 \to U(1) \to Spin^c(n) \to SO(n) \to 1 $$ Let $M$ be an oriented $n$-manifold and let $\pi \colon P \to M$ be the oriented frame bundle. This is a principal $SO(n)$-bundle. Let $$ P^{[2]} = \{ (p_1,p_2) \in P^2 \ | \ \pi(p_1) = \pi(p_2) \} $$ Then we have a principal $U(1)$-bundle obtained as follows: $$ L = \{ (p_1,p_2,g) \in P^{[2]} \times Spin^c\ | \ p_1\cdot q(g) = p_2 \} $$ where $q \colon Spin^c \to SO(n)$ is the canonical homomorphism. If we identify this $U(1)$-bundle with the associated line bundle, we have the following isomorphism $$ \mu \colon \pi_{12}^*L \otimes \pi_{23}^*L \to \pi_{13}^*L $$ where $P^{[3]}$ is defined analogously to $P^{[2]}$ and $\pi_{ij} \colon P^{[3]} \to P^{[2]}$ is the projection to the $i$th and $j$th factor. This should be thought of as a multiplication and satisfies an associativity constraint over $P^{[4]}$. This structure is called the lifting bundle gerbe.

How do you get the class $W_3(M)$ from this?

Choose a open cover $U_i$ of $M$, such that there are sections $\kappa_i \colon U_i \to P$, let $$ L_{ij} = (\kappa_i,\kappa_j)^*L . $$ Choose trivialisations $\theta_{ij} \colon U_{ij} \times \mathbb{C} \to L_{ij}$ (we can always choose the cover, such that these exist), where $U_{ij} = U_i \cap U_j$. Over the triple intersections $U_{ijk}$ we now have two trivialisations of $L_{ik}$. One is $\theta_{ik} $, the other one is $$ \mu_{ijk} \circ (\theta_{ij} \otimes \theta_{jk}) $$ where $\mu_{ijk}$ is the isomorphism $L_{ij} \otimes L_{jk} \to L_{ik}$ induced by $\mu$. The trivialisations differ by a continuous map $$ \omega_{ijk} \colon U_{ijk} \to U(1) $$ which turns out to be a Cech $2$-cocycle due to the associativity of the multiplication $\mu$. Hence, it represents an element $$ [\omega] \in \check{H}^2(M,U(1)) \cong H^3(M,\mathbb{Z}) . $$ This is the Spin$^c$ obstruction class.

  • $\begingroup$ There is also a direct connection with bundles of compact operators via groupoid C*-algebras .. $\endgroup$ Jan 13, 2017 at 18:07
  • $\begingroup$ Thank you for such great answer. Could you please give some references? For example where I can find the proof that $\beta(w_2(M))=\delta(M)$ (Dixmier Doaudy class) or details/proofs about the geometric construction which you described? $\endgroup$
    – truebaran
    Jan 14, 2017 at 13:42
  • $\begingroup$ @truebaran A good starting point to read about lifting bundle gerbes is Murray's paper "An introduction to bundle gerbes". Section 6.1 contains the things about the obstruction class I mentioned. $\endgroup$ Jan 15, 2017 at 19:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.