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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.

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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.

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  • $\begingroup$ There is also a direct connection with bundles of compact operators via groupoid C*-algebras .. $\endgroup$ – Ulrich Pennig Jan 13 '17 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 '17 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. $\endgroup$ – Ulrich Pennig Jan 15 '17 at 19:50

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