Dehn-Sommerville Theme

Dehn-Sommerville identities are symmetries for manifolds. First detected by Max Dehn in small dimensions, they were generalized by Duncan Sommerville in 1927. The relations were classically studied for simplicial polytopes which means q-spheres. It gives relations for the f-vector of a simplicial complex that is a q-sphere. The original work is messy. No wonder that the topic has been picked up again only with Victor Klee in 1963. It is now part of the “convex polytope” literature. A standard formulation is as follows: assume that G is a finite abstract simplicial complex that is a q-sphere and if is the f-vector of G, then half of these data are redundant the reason being that for all k=-1, …,q. This is quite ugly, isn’t it? It only looks so. Classically the topic has been tamed by a coordinate change and showing that it is equivalent that a h-vector is palindromic. This brings it close to Poincare-duality. I noticed other connections, especially [https://arxiv.org/abs/1905.04831: 2019: Dehn-Sommerville from Gauss-Bonnet] The funny thing is that I had never been looking for Dehn-Sommerville. It was just that whenever I researched a topic, Dehn-Sommerville came in. There is a probabilitistic interpretation using integral geometry, then there is a valuation approach using Gauss-Bonnet for valuations, then there is a Barycentric picture relating to eigenvectors of the Barycentric refinement operator [https://arxiv.org/abs/1509.06092: 2015: Universality for Barycentric subdivision] or [https://people.math.harvard.edu/~knill/graphgeometry/papers/invariants.pdf: 2015: Barycentric Characteristic numbers]. Finally there is a functional (calculus) approach using simplex generating functions [https://arxiv.org/abs/1905.04831: 2019: Dehn-Sommerville from Gauss-Bonnet]. There are other relations like connection calculus [https://arxiv.org/abs/1612.08229: 2016: On Fredholm determinants in topology], and Green functions [https://arxiv.org/abs/2010.09152: 2020: Green functions of Energized complexes] or [https://arxiv.org/abs/2302.02510: 2023 characteristic topological invariants] which are all very general. This all indicates that Dehn-Sommerville symmetries are at the heart of geometry. I started to review a bit the topic, after seeing that Dehn-Sommerville manifolds are actually quite a natural frame work in which one can study geodesic flows or interacting geodesics. [https://arxiv.org/abs/2503.18299, 2025: geodesics for discrete manifolds] and [https://www.arxiv.org/abs/2506.12054: 2025: interacting geodesics in discrete manifolds. I was led to the topic several times in my work on discrete geometry without knowing about the identities. I had worked on Gauss-Bonnet in 2009-2011 (see https://arxiv.org/abs/1111.5395) but could not explain then why odd dimensional q-manifolds had constant 0 curvature. That was resolved later using integral geometry: curvature is just an expectation of Poincare-Hopf indices and by pairing the indices of a function f and -f one gets a symmetric index j whose expectation is still curvature but which is zero for odd-dimensional manifolds simply because it is half the Euler characteristic of a level surface in the unit sphere of a point! Since the unit sphere is a q-1 sphere and a level surface is a (q-2) manifold by the discrete Sard theorem, we have constant zero curvature. Here are some links of work which stretches over more than a decade:[https://arxiv.org/abs/1201.1162: (2012: A graph theoretical Poincare-Hopf theorem], [https://arxiv.org/abs/1911.04208: 2019 Poincare Hopf for vector fields on graphs],[https://arxiv.org/abs/1202.4514 (2012 index expectation and curvature], [https://arxiv.org/abs/1205.0306 (2012: an index formula for simple graphs)], [https://arxiv.org/abs/1410.1217 (2014: Curvature from Graph Colorings], [https://arxiv.org/abs/1508.05657, 2015 A Sard theorem for graph theory],[https://arxiv.org/abs/2312.14671: 2023 Discrete algebraic sets in discrete manifolds]. While reviewing the subject, I decided to look more generally at Dehn-Sommerville manifolds and not only Dehn-Sommerville spheres. What happens is that just one of the invariants can drop which is the Euler characteristic in the even dimensional cases. The other identities still hold. There are two ways to see this: first of all the curvature of the valuation is just evaluated to the unit sphere. The second elegant Gauss-Bonnet formula is and realizing that the Dehn-Sommerville identities are equivalent to the fact that is either even or odd with respect to the point . One of the main reasons, why I like Dehn-Sommerville manifolds is Occam’s razor: if a geometric construct is simpler it might also more relevant. I would not be surprised, if Dehn-Sommerville manifolds would have an appearance in physics.

[Side remark: The later statement about simplicity is just for me an irrational guiding rule which I follow, similarly like the Church-Turing thesis, the law of small numbers or the assumption that we are able to think coherently, especially that our memory works (there is no puppet master or game designer or architect that can mess with our thinking). I explained a bit about these believes in this talk about odd perfect numbers, where I also explain a maybe more controversial meta principle: “if something is true there must be a reason why it is true” which might appear at first a bit silly but which I found to be a very good “gut feeling” principle to follow, especially if somebody claims something outrageous like having proven the Collatz conjecture or any of the https://www.youtube.com/watch?v=FENs3B7sxno major open problems in number theory. A proof needs to give insight which goes beyond the statement. It is for me the reason that I would bet strongly for the Collatz conjecture, the Riemann hypothesis to be false and that there are odd perfect numbers. The Dehn-Sommerville theme showed me that some really ugly identities are actually very beautiful if one looks at them in the right way! They are valuations for which the curvature is the same evaluation just one dimension lower. It confirms the irrational believe that beauty is hidden in good mathematics. It is (as I explained in that talk) also just a principle which I use to pick mathematical topics for my research. Mathematics (unlike other fields) is a like a candy store in the sense that there are billions of unexplored open problems and subjects. There is absolutely no problem to find something to work on. The problem is to SELECT something ! And for me, the selection criterion is “how beautiful it might be. Like in art, there is no economic “value functional” which can capture this. Beauty is something which is subjective. Some like complexity. I don’t. The original Dehn-Sommerville story as explained in textbooks or wikipedia is juist ugly! It is too complex. Not clear at all. Not general. I like “simplicity, generality and clarity” and this had been one of the epiphanies when starting to work on a Unix machine. If you compared the beauty of Unix with the early Macintosh systems (or heaven forbid DOS). Unix was an enlightement as well as a metaphor for “good taste”. And “good taste” is most important, but hard to define in mathematics. ]

The recording below was done at home because the 4th floor of the science center (including my office) goes through some renovations theses weeks. To clear up the shelves, I got rid of about 100 books and put up a temporary bookshelve while they work on one side of the office. I will see in a week or two how things will look. I decided to record outside on a small chalk board (which was maybe not a good idea outside in the sun) and recorded up some notes on the remarkable tablet. But it reminded me a bit about about the pandemic times. At the same spot, I had in 2021 (before chat GPT!) allowed my AI bots to teaching a math 1a class. The pandemic had been a nice (forced) opportunity to use technology. I had taught mostly from ipad tablets online, but had experimented also with real paper, like here (a lecture on improper integrals in 2021). The remarkable is surprisingly close to paper. In retrospect, the experimentation with technology was a positive part of that nightmare. It was important to keep the spirits up. (Here is an example of a lecture taught in Manga style).