With Urs Schreiber and Alessandro Valentino we are finalizing a short note on central extensions of mapping class groups from characteristic classes.

A preview of the note is available here: *Higher extensions of diffeomorphism groups (schreiber)*

Any comment or criticism is most welcome

]]>An extensive person entry Victor Snaith having some description of his 1979 Memoirs article (with a link to extended Russian translation). One of the aspects is an early version of an algebraic cobordism theory there, quite different both from Levine-Morel and from Voevodsky approach to alegbraic cobordism, and in a way to large. But the same construction is a special case of a more general metod in stable homotopy theory which may be of interest. Few words at algebraic cobordism as well. It would be nice to have more on Conner-Floyd isomorphism.

]]>Hi all,

sorry for having been absent lately, I’ve been fully absorbed by the preprint with Jim and Urs. now that’s over and my mind seems to be able again to follow nforum discussions :)

I’ve been thinking of oo-Chern-Simons theory. At present we are presenting it in nLab as a morphism $\mathbf{H}(\Sigma,A_{conn})\to \mathbf{B}^{n-dim \Sigma}U(1)$. This is fine but does not make explicit an important point: the relation to extended cobordism.

Let me sketch it (in a simplified situation where I will not consider differential refinements). We have a cocycle $c:A\to \mathbf{B}^n U(1)$. if we now consider an $n$-representation of $\mathbf{B}^n U(1)$, e.g., the fundamental one, then we can see $c$ as the datum of an $n$-vector bundle over $A$. Now, it is likely that $n$Vect$(A)$ is a symmetric monoidal $(\infty,n)$-category, and it is hopeful that the $n$-vector bundle corresponding to $c$ is a fuly dualizable object. So by the cobordism hypothesis we get a representation of $Bord_n$ with values in $n$Vect$(A)$. In particular to a closed connected $n$-manifold $\Sigma$ it will correspond a $0$-vector bundle over $A$, i.e. a complex valued function on $A$ constant over the isomorphism classes of objects. Integrating this over $A$ produces the invariant associated to $\Sigma$.

We can associate a cobordism invariant to $\Sigma$ also in another way: first we push the given $n$-vector bundle forward to the point, i.e. we take the $n$-vector space of its sections. This is hopefully fully dualizable, so we have a representation $Bord_n\to n$Vect. And so an invariant associated to $\Sigma$. It is reasonable to expect that these two invariants are the same.

There is one more point of view on this: namely, we can consider bordism with values in $A$. if a representation $Bord_n(A)\to n$Vect is given, then to a morphism $c:\Sigma\to A$ will correspond a $(n-dim\Sigma)$-vector space $V_c$. this gives a $(n-dim\Sigma)$-vector bundle over $\mathbf{H}(\Sigma,A)$. if the $n$-vector space $V_c$ is the fundamental representation of $\mathbf{B}^{n-dim\Sigma} U(1)$, then the $(n-dim\Sigma)$-vector bundle over $\mathbf{H}(\Sigma,A)$ is induced by a principal $\mathbf{B}^{n-dim\Sigma} U(1)$-bundle, i.e., it corresponds to a morphism $\mathbf{H}(\Sigma,A)\to \mathbf{B}^{n-dim\Sigma} U(1)$, which is where we started from.

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