In our *Mapping surgery to analysis* papers, Nigel and I proposed an analytic counterpart of the *surgery exact sequence* which summarizes the main results of the (Browder, Novikov, Sullivan, Wall) theory of high-dimensional manifolds. This exact sequence identifies the set of manifold structures within a given homotopy type \(X\) (the *structure set*) as the fiber of an assembly map

\[ H_*(X; {\mathbb L}(e)) \to L_*({\mathbb Z}\pi_1(X)) \]

which abstracts the ides of obtaining “signature obstructions” from a “surgery problem”.

Analogously, we constructed an *analytic structure set* (actually the K-theory of a certain C*-algebra) as the fiber of a Baum-Connes type assembly map, and showed that index theory provides a natural transformation from the topological surgery exact sequence to our *analytic surgery exact sequence*.

Our structure set is defined in purely analytic terms. However, in a subsequent paper where we related our exact sequence to the theory of \(\eta\) invariants, it became useful to have a more geometrical approach to the structure set also. (The relation between the “more geometrical” and “more analytical” approaches is roughly the same as that between the Baum-Douglas and Kasparov models of K-homology.) Our paper didn’t give a geometric *definition* of the structure set – just a geometric *approach* to certain elements.

A recent arXiv paper by Deeley and Goffeng proposes to take this idea to its logical conclusion by constructing a Baum-Douglas type model for the whole analytic structure set. The basic idea is this: An element of the structure set should be “an elliptic operator together with a reason that its index vanishes”. The cobordism invariance of the index shows that one example of such a “reason” is that our elliptic operator is actually defined on the *boundary* of some manifold (and that our operator is a boundary operator). Therefore a first approximation to a Baum-Douglas model of the structure set should have as cycles spin-c manifolds with boundary \( (M,\partial M) \) together with maps \(\partial M \to X\).

But of course this (cobordism) is not the *only* known reason for the vanishing of an index (e.g., as I understand it, the fundamental question about positive scalar curvature metrics is whether positive scalar curvature *implies* some bordism condition). So suppose you have an elliptic operator whose index vanishes for some “positive scalar curvature type” reason. How are you to build a structure class? It seems to me that Deeley-Goffeng deal with this by incorporating quite a lot of analysis into their geometric cycles – as well as the bordism that I have described, there are also projective module bundles over the group algebra, etc… this makes the desired exactness true, but perhaps at the cost of making the groups less geometrical; they are a “geometry-analysis hybrid”. And that is inevitable in this problem.

I should mention that several other applications of the analytic surgery sequence depend on constructing an appropriate ncie model for the structure set: e.g. Siegel, Xie-Yu (see below). I’m not sure whether our original model is “nice” for anybody!

#### References

Deeley, Robin, and Magnus Goffeng. *Realizing the Analytic Surgery Group of Higson and Roe Geometrically, Part I: The Geometric Model*. ArXiv e-print, August 27, 2013. http://arxiv.org/abs/1308.5990.

Higson, Nigel, and John Roe. “Mapping Surgery to Analysis. I. Analytic Signatures.” *K-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of K-Theory in the Mathematical Sciences* 33, no. 4 (2005): 277–299. doi:10.1007/s10977-005-1561-8.

———. “Mapping Surgery to Analysis. II. Geometric Signatures.” *K-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of K-Theory in the Mathematical Sciences* 33, no. 4 (2005): 301–324. doi:10.1007/s10977-005-1559-2.

———. “Mapping Surgery to Analysis. III. Exact Sequences.” *K-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of K-Theory in the Mathematical Sciences* 33, no. 4 (2005): 325–346. doi:10.1007/s10977-005-1554-7.

Higson, Nigel, and John Roe. “\(K\)-homology, Assembly and Rigidity Theorems for Relative Eta Invariants.” *Pure and Applied Mathematics Quarterly* 6, no. 2, Special Issue: In honor of Michael Atiyah and Isadore Singer (2010): 555–601.

Siegel, Paul. “The Mayer-Vietoris Sequence for the Analytic Structure Group.” *arXiv:1212.0241* (December 2, 2012). http://arxiv.org/abs/1212.0241.

Siegel, Paul. “Homological Calculations with the Analytic Structure Group.” PhD Thesis, Penn State, 2012. https://etda.libraries.psu.edu/paper/16113/.

Xie, Zhizhang, and Guoliang Yu. “A Relative Higher Index Theorem, Diffeomorphisms and Positive Scalar Curvature.” *arXiv:1204.3664* (April 16, 2012). http://arxiv.org/abs/1204.3664.

Xie, Zhizhang, and Guoliang Yu. “Positive Scalar Curvature, Higher Rho Invariants and Localization Algebras.” *arXiv:1302.4418* (February 18, 2013). http://arxiv.org/abs/1302.4418.