Dehn Sommerville Mindmap - Quantum Calculus

Next week, I will be back in my office. As they are constructing labs just near my temporay office, I made this “talk to myself session” in a seminar room of the department. I try to finish these days a review about Dehn-Sommerville, a rather unpopular topic historically speaking (not even one mathematician seems have been interested in this now 100 year old topic until Victor Klee picked it up in the sixties). One can understand this. The original paper of Sommerville is almost unreadable and also in current combinatorial topology books about polyhedra, the topic is ugly. I still think today, that my approach from 2019 nailed it: instead of looking at coefficients of the f-vector, one has to look at generating functions, the simplex generating function. I shifted my definition a bit and introduced now Dehn-Sommerville manifolds which are spaces for which every unit sphere is a Dehn-Sommerville sphere, a Dehn-Sommerville manifold with the Euler characteristic of a sphere of the same dimension. It looks as if everybody looked at Dehn-Sommerville only for spheres (like when looking at convex polytopes). I have not yet seen that anybody has realized before my Barycentric characteristic numbers paper, that for example that the $f$ -vector of every 4-manifold satisfies $-22 f_1 + 33 f_2 -40 f_3 + 45 f_4=0$ . It is not only for spheres, it also works for any 4-manifold, even if it does not satisfy Poincare duality and as I realized also for recursively defined spaces: $\mathcal{D}_q=\{ G, \forall x \in G, S(x) \in \mathcal{D}_{q-1}, \chi(S(x))=1-(-1)^q \}$ with the induction assumption that $\mathcal{D}_{-1}=\{ \emptyset \}$ contains only the void the empty simplicial complex.

EXAMPLE 1: Take for example the K3 surface G, one of the most famous 4-manifolds (it is called surface because it is a complex surface). It can be seen as a quartic surface $x^4+y^4+z^4+w^4=0$ in the complex 3-manifold $\mathbb{CP}^3$ (which is a real 6-manifold). There is an small implementation with f-vector $f = (16, 120, 560, 720, 288)$ . If you take the dot product with (0, -22, 33, -40, 45) (one of the eigenvectors of the Barycentric refinement operator in 4 dimensions) you get 0. Now, the K3 surface satisfies Poincare duality as it is orientable (its Betti vector is (1,0,22,0,1)).

EXAMPLE 2: So , lets look at an other 4-manifold, the real projective space $\mathbb{RP}^4$ which is a nice positive curvature 4-manifold. There is a small implementation with $f = (16, 120, 330, 375, 150)$ . Again this is perpendicular to the Barycentric Dehn-Sommerville eigenvector (0, -22, 33, -40, 45). Now, the $\mathbb{R}P^4$ manifold is not orientable and does not satisfy Poincar\’e duality. Its Betti vector is (1,0,0,0,0).

EXAMPLE 3. Let G be a twisted circle bundle over the 3-sphere. While the Betti vector of the product $S^3 \times S^1$ is (1,1,0,1,1) and satisfies Poincare duality, the twisted version does have Betti vector (1,1,0,0,0). It is not orientable. There is a small implementation with f-vector $f=(12, 60, 120, 120, 48)$ . Again $f.(0,-22,33,-40,45) = 0$ .

EXAMPLE 4: Let G be the complex projective plane $CP^2$ . It has Betti vector b=(1,0,1,0,1). There is a small implementation with f-vector $f=(9, 36, 84, 90, 36). It satisfies naturally the Dehn-Sommerville symmetry$ latex f.(0,-22,33,-40,45) = 0$.

EXAMPLE 5: Lets take an example of a suspension of the 3-manifold $M=mathbb{R}P^3$. A suspension of a 3-manifold M is only a manifold if M is a sphere. Since $M=mathbb{R}P^3$ is not a sphere, $G=M \oplus 2=M \oplus S^0$ is no more a manifold. But it is a Dehn-Sommerville 4-manifold. The Betti vector of G is the same than for a 4-sphere, but it is not a 4-sphere. There is a small implementation of that suspension with f-vector $(13, 73, 182, 200, 80)$ . Of course, it is satisfies the Dehn-Sommerville symmetry.

EXAMPLE 6: Now start with the 4-manifold from before. Since it has Euler characteristic 2, it actually is a Dehn-Sommerville 4-sphere! Now take the suspension again to get a 5-manifold. It is the double suspension of the projective space $RP^3 and not a manifold. It can also be seen as the join of$ latex RP^3$ with the 2-sphere $S^2$ . The f-vector of a small implementation of this Dehn-Sommerville 5-manifold is (15, 99, 328, 564, 480, 160). As a 5-manifold this f-vector is perpendicular to the Euler characteristic vector (1,-1,1,-1,1,-1). Beside the other “trivial” Dehn Sommerville vector (0,0,0,0,-1,3), we have the non-trivial Dehn-Sommerville symmetry given by (0,0,-19,38,-55,70) in dimension 5. And of course (15, 99, 328, 564, 480, 160) . (0, 0, -19, 38, -55, 70) = 0.

EXAMPLE 7: Lets look at the suspension of the K3 surface. It is a 5 dimensional simplicial complex. There is a small implementation with f-vector $f=( 18, 152, 800, 1840, 1728, 576)$ . Now, (18, 152, 800, 1840, 1728, 576) . (0, 0, -19, 38, -55, 70) = 0. But this was just luck because the suspension of the K3 surface is not 5 manifold and not a Dehn-Sommerville 5 manifold. Indeed, the Euler characteristic of this 5-manifold is 2-24=-22 and not 0 as it should be for any 5-Dehn-Sommerville manifold. But the other Dehn-Sommerville symmetries still hold.

EXAMPLE 8: If we look at the join of $T^2$ and $S^2$ we get a 5 dimensional complex. A small implementation has f-vector (14, 84, 264, 448, 384, 128). It is no more perpendicular to (0, 0, -19, 38, -55, 70). It did not help that $S^2$ was a Dehn-Sommerville sphere. This already fails in smaller dimension. The join of $T^2$ with $S^1$ is a 4-manifold which has an f-vector not perpendicular to (0,-22,33,-40,45).

EXMPLE 9: Lets look at a the suspension of a disjoint union of a 2-torus and a 2-sphere. This is a Dehn-Sommerville 4-manifold but not a 4-manifold. There is a small implementation with f=(18, 96, 224, 240, 96). This is indeed perpendicular to (0, -22, 33, -40, 45).

EXAMPLE 10: Lets look at the double suspension of a homology 3-sphere. It is a Dehn-Sommerville 5-manifold but not a 5 manifold. This is an interesting example because its geometric realization is actually (classically) homeomorphic to the 5 sphere by the double suspension theorem (of course it is not in the discrete). There is a small implementation of G with f-vector (28, 254, 972, 1786, 1560, 520). This is perpendicular to the Barycentric Dehn-Sommerville vector (0, 0, -19, 38, -55, 70). The other invariant is Euler characteristic which of course is zero (as an odd dimensional manifold) and the trivial valuation (0,0,0,0,-2,6) for a 5-manifold (which by the way is the only valuation we need to have vanished in order to define a geodesic flow)

The topic is related to many interesting and elegant results like Gauss-Bonnet, Poincare-Hopf, connection calculus, Green function identities, valuations, Barycentric refinement … and last when looking at geodesic flows. I tried again, like last week on my little board at home to make a mind-map. I should mention an other shift in language since my 2019 paper. That paper was written in the language of graphs which is of course almost the same as graphs define simplicial complexes and simplicial complexes describe graphs. There is one advantage when talking about simplicial complexes in that there is less misunderstanding in that mathematicians think that it is about one dimensional spaces (it is a rather entrenched point of view to see graphs as one dimensional simplicial complexes). With simplicial complexes there is an other source of misconception however (which is also historical, even in the original Dehn-Heegard article introducing abstract simplicial complexes geometric realizations were mixed in) that one thinks about geometric realizations. A simplicial complex is just a finite set of sets. No infinity axiom is invoked ever. Infinity can stay in hell where it might actually belong to as it is perfectly possible that any mathematical construct invoking infinity is inconsistent. And because we can never prove consistency anyway, why bother at all with getting into that mess? Geometry in the finite is so much simpler.

Here is an other mind map done when realizing that the board is not that well visible.

Author: oliverknill