In this short paper, posted in july 2003, G. Perelman shows that for any closed oriented three-manifold M$M$, whose prime decomposition contains no aspherical factors, the Ricci flow with surgery stops in finite time. To stop means that the scalar curvature becomes large everywhere on M$M$ so that it is covered by canonical neighbourhoods and hence the topology is completely known; it is also said that the flow becomes extinct in finite time. It is an important step in the proof of the Poincaré conjecture; indeed it is a short cut which avoids using the long time behaviour of the Ricci flow described in [G. Perelman, “Ricci flow with surgery on three-manifolds”, arXiv: math.DG/0303109 (2003; Zbl 1130.53002)] sections 6 to 8.
The idea is to fill some suitably chosen loop by a minimal disc and let both the metric evolve by the Ricci flow and the loop by the curve shortening flow. This idea is similar to one used by R. Hamilton in [R. Hamilton, Commun. Anal. Geom. 7, No. 4, 695–729 (1999; Zbl 0939.53024)] and a very detailed proof is given in [J. Morgan and G. Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs 3 (2007; Zbl 1179.57045)]. An alternative approach using harmonic maps is given in [T. Colding and W. Minicozzi, J. Am. Math. Soc. 18, No. 3, 561–569 (2005; Zbl 1083.53058)] and with more details in [T. Colding and W. Minicozzi, “Width and finite extinction time of Ricci flow”, arXiv:0707.0108].

This is the second paper written by G. Perelman which proves the geometrization conjecture. It is presented by the author as a “technical paper” in which some assertions made in [G. Perelman, “The entropy formula for the Ricci flow and its geometric applications”, arXiv:math.DG/0211159 (opens in new tab) (2002; Zbl 1130.53001)] are proved and some others are corrected. It presents the Ricci flow with surgery on three-manifolds which is a version of the Ricci flow taking into account the singularities. It is inspired by the construction made by R. Hamilton in [Commun. Anal. Geom. 5, No. 1, 1–92 (1997; Zbl 0892.53018)]. As for G. Perelman [loc. cit.] the details can be found in [B. Kleiner and J. Lott, “Notes on Perelman’s papers”, arXiv:math.DG/0605667 (opens in new tab) (2006); J. Morgan and G. Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs 3 (2007; Zbl 1179.57045) and H.-D. Cao and X.-P. Zhu, Asian J. Math. 10, No. 2, 165–492 (2006; Zbl 1200.53057)].
Sections 1 and 2: These sections present preliminary material on ancient and standard solutions. In section 1 the author gives a more precise classification of ancient solutions by the study of their asymptotic soliton. Let us recall that the ancient solutions are those which have an infinite past and ought to be (with some more properties) the infinitesimal models for the singularities. The asymptotic soliton is a limit when time goes to −∞$-\mathrm{\infty }$ of a rescaling of the ancient solution. These notions are defined in section 11 of Perelman [loc. cit.] and more refined properties are described here.
Section 2 is devoted to the standard solution. This is a solution of the Ricci flow whose initial data is a half cylinder capped-off by a hemisphere. The initial data being non compact existence and uniqueness of the solution is not immediate. It is shown that the Ricci flow with such an initial data exists in the time interval [0,1)$[0,1)$ and is unique. An extremely detailed proof is given in [Morgan and Tian, loc. cit.; see also Kleiner and Lott, loc. cit., and Cao and Zhu, loc. cit.].
Section 3: In this important section the first time for which the flow becomes singular is described. Being singular means that the scalar curvature becomes infinite somewhere. If it is infinite everywhere then the manifold is covered by canonical neighbourhoods and its topology is thus completely known. If it becomes infinite on a subset Ω$\mathrm{\Omega }$, it is open and one can describe the part of Ω$\mathrm{\Omega }$ of high scalar curvature. They are, again, covered with canonical neighbourhoods. This leads to the important notion of horns which are subsets of Ω$\mathrm{\Omega }$ diffeomorphic to open half cylinders and on which the scalar curvature goes to infinity on one end. It is in these horns that the surgery will take place. The subset M∖Ω$M\setminus \mathrm{\Omega }$ is covered by canonical neighbourhoods and is topologically simple (and known).
Sections 4 and 5: These are the sections in which the Ricci flow with surgery is defined. In section 4 the so-called Ricci flow with cutoff appears. It is given by a collection of smooth Ricci flows defined on a 3-manifold on adjacent intervals of time. On the boundary of these intervals surgeries take place. In this section it is supposed that the flow satisfies the a priori assumptions which are: the Hamilton-Ivey pinching property and the canonical neighbourhood property for points whose scalar curvature is larger than a parameter r>0$r>0$. The surgery is done in horns. More precisely it is shown that if one goes “far enough” into a horn then one finds a neck of size 2/δ$2/\delta$ for some parameter δ>0$\delta >0$ small enough and of curvature close to h>0$h>0$ another parameter depending on r$r$ and δ$\delta$. The flow is called a Ricci flow with δ$\delta$-cutoff. Now, this δ$\delta$-neck is cut in the middle and a suitably rescaled compact piece of the standard solution is glued on one side. The flow may start again with the new Riemannian manifold thus obtained. An important property is the fact that the added piece (called an almost-standard cap) remains close to the evoluting standard solution for a while.
In section 5 the author shows that the flow can be defined, satisfying the a priori assumptions for all time. The parameters r$r$ and δ$\delta$ must now depend on the time parameter and one issue is to prove that they do not go to zero in finite time. The proof is very close to the proof of the canonical neighbourhood theorem done in [Perelman, loc. cit.] section 12, taking into account the surgeries. This is the key result of this series of work by G. Perelman.
Sections 6 and 7: Now, the long time behaviour of the Ricci flow with surgery is analysed carefully. Section 6 presents technical issues such as curvature estimates in the future and the past of a given time slice.
Section 7 defines and describes the thick-thin decomposition of the manifold. It is inspired by R. Hamilton [Commun. Anal. Geom. 7, No. 4, 695–729 (1999; Zbl 0939.53024)]. It is shown that thick parts become more and more hyperbolic bounded by incompressible tori. The description of the thin part relies on an unpublished paper by the author; the conclusion is that it is a graph manifold.
Section 8: It contains an alternative approach. The geometric decomposition is described using the values of a Riemannian invariant as threshold for the different possibilities. This invariant is the first eigenvalue of a Schrödinger operator whose potential is given by the scalar curvature. A simpler argument can be found in [Kleiner and Lott, loc. cit., Section 93].
At the time when this review is written the fact that the thin part is a graph manifold is still a bit controversial although it is widely believed to be true. Some details are missing in the literature. An alternative approach is given in [L. Bessières, G. Besson, M. Boileau, S. Maillot and J. Porti, “Suites de métriques extraites du flot de Ricci sur les variétés asphériques de dimension 3”, arXiv:0706.2065 (opens in new tab)].

This is the first part of a masterpiece of mathematics which leads to a proof of the Poincaré and geometrization conjectures. Although the most striking application is in dimension 3 a good deal of the results presented here are valid in an arbitrary dimension n$n$. This is the case for sections 1 to 10 of the present paper. It concerns the Ricci flow introduced by Richard Hamilton in the celebrated paper [R. Hamilton, J. Differ. Geom. 17, 255–306 (1982; Zbl 0504.53034)]. Brief descriptions of each section follow. The detailed proofs can be read in [B. Kleiner and J. Lott, “Notes on Perelman’s papers”, arXiv:math.DG/0605667 (opens in new tab) (2006)], [J. Morgan and G. Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs 3 (2007; Zbl 1179.57045)] and [H.-D. Cao and X.-P. Zhu, Asian J. Math. 10, No. 2, 165–492 (2006; Zbl 1200.53057)].
Section 1: On a closed manifold M$M$ the Ricci flow is presented as a gradient flow of a functional defined on the couples (g,f)$(g,f)$, where g$g$ is a Riemannian metric and f$f$ is a function on M$M$. the function is chosen so that e−fdvolg$e^{-f}d{\mathrm{vol}}_g$ is a fixed measure. The functional is in which R$R$ denotes the scalar curvature of the metric g$g$ and the norm is taken for this metric.
Section 2: This section and the following one present immediate corollaries which we can summarize as follows: On the space obtained by moding out the space of metrics by the action of the diffeomorphisms group and the homotheties there are no closed trajectories of the Ricci flow. In Perelman’s terminology one shows that there are no non trivial breathers. This section is devoted to the proof of this assertion in the case of steady and expanding breathers. This relies on introducing very interesting invariants which are nondecreasing along the flow and stationary on Ricci solitons. One is the infinum on f$f$ of F(g,f)$\mathcal{F}(g,f)$ which turns out to be the smallest eigenvalue of −4Δ+R$-4\mathrm{\Delta }+R$. The other one is the same eigenvalue normalised by the volume raised at the suitable power in order to get a scale invariant quantity.
Section 3: This section deals with the shrinking breathers which are more difficult to study. A modification of F$\mathcal{F}$ called W$\mathcal{W}$ is introduced. It is a function of three parameters: a metric g$g$, a function f$f$ and a number τ$\tau$ which is t0−t$t_0-t$ where t$t$ is the parameter of the Ricci flow. Again W$\mathcal{W}$ is nondecreasing along the Ricci flow when f$f$ is normalized as in section 1 and is stationary on gradient shrinking solitons.
Section 4: This is the second breakthrough of this paper. It presents a tool that gives its full strength to the Ricci flow, namely a solution on a closed manifold M$M$ is non collapsed at a finite time t$t$. It means there exists a number κ>0$\kappa >0$ such that balls at time t$t$ and of radius r≤ρ$r\le \rho$ on which the Riemann curvature is bounded by r−2$r^{-2}$ have volume bounded below by κrn$\kappa r^n$. The number , called the scale, is the value below which this property is true. All these quantities are scaled invariant which makes this property true even after rescaling the metrics. This implies a local injectivity radius bound and allows to apply compactness theorems. This is crucial for the singularities analysis.
Section 5 and 6: These two sections are less useful at the moment. They intend to give justifications and explanations for the previous construction either using a statistical mechanics’ approach or the point of view of infinite dimensional geometry.
Section 7: In this very important section the author develops the Morse theory of a functional called L$\mathcal{L}$-length, which is a space-time version of the standard length (or energy). L$\mathcal{L}$-geodesics, L$\mathcal{L}$-Jacobi fields and L$\mathcal{L}$-exponential are computed leading to some comparison theorems. The reduced volume is introduced, a monotonic quantity along the Ricci flow which allows to give an alternative proof of the nonlocal collapsing property. This is a key section for the three dimensional applications of the Ricci flow.
Section 8: A refined version of the nonlocal collapsing theorem is proved using the tools developed in the previous section, in particular the reduced volume.
Section 9: A differential Harnack inequality is proved.
Section 10: In this section it is proved that despite the fact that the Ricci flow is not local, it is pseudo-local. Namely, if the curvature is close to zero in a region (and with extra assumption), at least for a short time it remains not too far from zero, i.e., for a short time it is not affected by the possible presence of big curvature elsewhere on the manifold. This fact is made precise.
Section 11: This is a section devoted to the classification of the so-called κ$\kappa$-solutions in dimension 3. They are solutions of the Ricci flow which are ancient (i.e., have an infinite past), non flat, have bounded and nonnegative curvature operator on each time slice and are κ$\kappa$-non-collapsed at all scales for a positive κ$\kappa$. They are infinitesimal models for the singularities of a three-dimensional Ricci flow. Indeed, in three dimensions, they are obtained as limits of suitable blow-up around a singularity (points of very large curvature) and at finite time. This classification is necessary in order to understand the Ricci flow with surgery.
Section 12: This section presents another breakthrough, the so-called canonical neighbourhood theorem. This is again specific to the three-dimensional case. It is shown that for a smooth Ricci flow, with normalised initial data, there is a universal number such that if at some point the scalar curvature becomes larger than this value, then a neighbourhood of this point, of controlled size, is, after rescaling, close to a piece of a κ$\kappa$-solution. This describes quite precisely the geometry of the manifold, whose evolution is given by the Ricci flow equation, around points of large curvature. The rest of the section is devoted to the (beginning of the) long term analysis of the Ricci flow.
Section 13: In this section the proof of geometrisation is sketched for manifolds which carry a Ricci flow defined for all time. It is shown that the manifold has a thick-thin decomposition, that the thick part becomes hyperbolic and is bounded by incompressible tori following arguments due to R. Hamilton. The thin part is claimed without proof to be a graph manifold. Some arguments which are adapted from the works of R. Hamilton are more precisely treated in [G. Perelman, “Ricci flow with surgery on three-manifolds”, arXiv:math.DG/0303109 (opens in new tab) (2003; Zbl 1130.53002)]. Others concerning the case where the flow develops singularities are sketched but not justified. The reader is referred to [Perelman, loc. cit.].

The author constructs an example of a complete Riemannian manifold of positive Ricci curvature with Euclidean volume growth and such that its asymptotic cone is not unique.
For the entire collection see [Zbl 0871.00023].

Author’s abstract: “It is shown that a connected sum of an arbitrary number of complex projective planes carries a metric of positive Ricci curvature with diameter one and, in contrast with the earlier examples of J.-P. Sha and D.-G. Yang [J. Differ. Geometry 29, 95-103 (1989; Zbl 0633.53064); ibid. 33, 127-137 (1991; Zbl 0728.53027)] and M. T. Anderson [Manuscr. Math. 68, 405-415 (1990; Zbl 0711.53036)], with volume bounded away from zero. The key step is to construct complete metrics of positive Ricci curvature on the punctured complex projective plane, which have uniform Euclidean volume growth and almost contain a line, thus showing topological instability of the splitting theorem of J. Cheeger and D. Gromoll [J. Differ. Geometry 6, 119-128 (1971; Zbl 0223.53033)], even in the presence of the lower volume bound. In the absence of such a bound, the topological instability was earlier shown by M. T. Anderson [Topology 29, 41-55 (1990; Zbl 0696.53027)]; metric stability holds, even without the volume bound, by the recent work of Colding and Cheeger”.
For the entire collection see [Zbl 0871.00023].

This is a technical paper devoted to the investigation of collapsing of Alexandrov spaces with lower curvature bound [for motivation for the collapsing problem, see J. Cheeger, K. Fukaya and M. Gromov, J. Am. Math. Soc. 5, 327-372 (1992; Zbl 0758.53022) and references therein]. In a previous paper [G. Ya. Perel’man, St. Petersbg. Math. J. 5, 205-213 (1994; Zbl 0815.53072)], the author defined a canonical stratification of an Alexandrov space by so-called extremal subsets. It is likely that if the limit of a collapsing sequence has no proper extremal subsets, then the collapsing spaces are fiber bundles over the limit space.
In this paper, a weaker statement is proved about the homotopy groups of those spaces. As a corollary, one deduces that the asymptotic cone of a complete noncompact Riemannian manifold of nonnegative sectional curvature, that does not admit isometric splitting and is not diffeomorphic to Rn${\mathbb{R}}^n$, has proper extremal subsets. In particular, using a result of the author and A. M. Petrunin [St. Petersbg. Math. J. 5, 215-227 (1994; Zbl 0802.53019)] it follows that the radius of its ideal boundary is at most π/2$\pi /2$. This last assertion was conjectured by T. Shioya [Math. Z. 212, 223-238 (1993; Zbl 0791.53045)].
For the entire collection see [Zbl 0871.00023].

In this short and concise survey article, the author gives an overview of some of his work on Alexandrov spaces of curvature bounded below, namely on elementary Morse theory on Alexandrov spaces and on extremal subsets of Alexandrov spaces. A discussion of open problems concludes the paper.
For the entire collection see [Zbl 0829.00014].

Let X$X$ be a metric space. The k$k$-dimensional (Uryson) width wk(X)$w_k(X)$ of X$X$ is defined as the exact lower bound of those δ>0$\delta >0$ for which there exists a k$k$-dimensional space P$P$ and a continuous map f:X→P$f:X\to P$ all of whose inverse images have diameters at most δ$\delta$. The author presents a proof of the following conjecture, formulated by M. Gromov [On the geometry of differentiable manifolds, Workshop, Rome/Italy 1986, Astérisque 163-164, 93-109 (1988; Zbl 0684.53036)]: The volume and the widths of a closed Riemannian manifold Mn$M^n$ with nonnegative sectional curvature satisfy where c=c(n)$c=c(n)$ is a constant. The proof consists of two steps.
First one introduces a different collection of measurements, called packing widths, and one proves the above inequalities for those. Then essential equivalence between widths and packing widths is established. The author works in the category of Alexandrov spaces which is largely explained by the fact that the natural process of taking Gromov-Hausdorff limits is closed in this category but not in the Riemannian category. The paper of Yu. Burago, M. Gromov and the author [Russ. Math. Surv. 47, No. 2, 1-58 (1992); translation from Usp. Mat. Nauk 47, No. 2(284), 3-51 (1992; Zbl 0802.53018)] is the basis of most of his arguments.

The author gives a short proof of the following theorem: Let n≥2$n\ge 2$ and k∈R$k\in \mathbb{R}$. Then there exists ε=ε(n,k)>0$\epsilon =\epsilon (n,k)>0$ so that any n$n$-dimensional closed Riemannian manifold with sectional curvature secM≥−k${\text{sec}}_M\ge -k$, Ricci curvature RicM≥n−i${\text{Ric}}_M\ge n-i$ and diameter diamM≥π−ε${\text{diam}}_M\ge \pi -\epsilon$ is a twisted sphere. The proof uses the argument from K. Grove and K. Shiohama [Ann. Math., II. Ser. 106, 201–211 (1977; Zbl 0341.53029)].

The soul conjecture of Cheeger and Gromoll states that the soul of a noncompact Riemannian manifold of nonnegative sectional curvature is a point if there is a point in the manifold where the sectional curvatures of all the 2-planes is positive. The author proves a more precise version of this theorem.

The main result of the paper is theorem 1: for any integer n≥2$n\ge 2$ there exists δn>0${\delta }_n>0$ with the following property. Let Mn$M^n$ be a complete Riemannian manifold with Ric≥n−1$\text{Ric}\ge n-1$. Suppose that Vol(Mn)≥(I−δn)Vol(Sn(1))$\text{Vol}(M^n)\ge (I-{\delta }_n)\text{Vol}(S^n(1))$. Then Mn$M^n$ is homeomorphic to Sn$S^n$. The proof is based on the well-known works of R. S. Hamilton [J. Differ. Geom. 17, 255-306 (1982; Zbl 0504.53034)], S. Smale [Ann. Math., II. Ser. 74, 391-406 (1961)] and M. H. Freedman [J. Differ. Geom. 17, 357-453 (1982; Zbl 0528.57011)]. The author proves that under the assumptions of theorem 1, πi(Mn)=0${\pi }_i(M^n)=0$, for all i<n$i<n$ and refers to the papers cited above. The vanishing of appropriate homotopy groups follows from a lemma about the extension of any continuous map f:Sk=∂Dk+1→Bp(R)$f:S^k=\mathrm{\partial }{D}^{k+1}\to B_p(R)$ to a map g:Dk+1→Bp(c1R)$g:D^{k+1}\to B_p(c_{1}R)$ and a continuous map f:Sk→M∖Bp(R)$f:S^k\to M\mathrm{\setminus }{B}_p(R)$ to a map into M∖Bp(c1R)$M\mathrm{\setminus }{B}_p(c_{1}R)$ (continuous deformation). Here c2>c1>1$c_2>c_1>1$, Z∋k≥0$\mathbb{Z}\ni k\ge 0$, 0<R<πc−12$0<R<\pi c_2^{-1}$, Bp(R)$B_p(R)$ is the ball of radius R$R$, p∈M$p\in M$ and Mn$M^n$ satisfied some inequality involving volumes of the balls in Mn$M^n$ and Sn(1)$S^n(1)$. The possibility of the extension and deformation is proved by induction on k$k$ because it appears to be possible to construct the appropriate homotopy on the (k−1)$(k-1)$-skeleton of the fine triangulation of Sk$S^k$. Some other homotopy results are given.

Authors’ abstract: Using the concept of an extremal subset, the authors construct a natural stratification of any Aleksandrov space. This stratification takes into account topological as well as metric properties of the space. Several properties of the stratification are established, the most important being the quasi-geodesicity of the strata. This generalizes the classical theorem of Liberman on geodesics on convex hypersurfaces in Rn${\mathbb{R}}^n$.

Summary: We present a version of elementary Morse theory for distance functions on an Aleksandrov space with curvature bounded from below. It is proved that 1) any small spherical neighborhood of a point in an Aleksandrov space is homeomorphic to the open cone over its boundary, and 2) any Aleksandrov space admits a canonical stratification whose strata are topological manifolds.

The theory of (basically finite-dimensional) metric spaces with curvature (in the sense of Alexandrov) bounded below is developed. Roughly speaking, it is dealing with spaces with an intrinsic metric, for which the conclusion of Toponogov’s angle comparison theorem is true (although only in the small). These spaces are defined axiomatically by their local geometric properties, without the techniques of analysis. They may have metric and topological singularities, in particular, they may not be manifolds. The class considered includes all limit spaces of sequences of complete Riemannian manifolds with sectional curvature uniformly bounded below. Alexandrov spaces arise naturally if Riemannian manifolds are considered from the viewpoint of synthetic geometry and one avoids the excessive assumptions of smoothness connected with the use of analytic techniques. Spaces with singularities may appear as limits of sequences of ordinary Riemannian manifolds, and therefore the first are necessary for studying the latter.

See the review in Zbl 0681.53032.

See the review in Zbl 0714.53035.

The author constructs an example of a complete analytic saddle surface in a 4-dimensional Euclidean space R4$R^4$ with the Gaussian curvature K≤−c2<0$K\le -c^2<0$, c=const$c=const$. The author determines the equation of the surface in the form r(x,y)=(x,y,α1(ζ),α2(ζ)),ζ=x+iy$\mathfrak{r}(x,y)=(x,y,{\alpha }_1(\zeta ),{\alpha }_2(\zeta )),\zeta =x+iy$, where x,y are coordinates in a 2-dimensional plane; α1,α2${\alpha }_1,{\alpha }_2$ are harmonic functions of x,y. The author also constructs a complete saddle surface in a 6-dimensional space R6$R^6$ with non-degenerate ellipses of normal curvatures and another one with K≤−c2<0$K\le -c^2<0$.

A metric on a surface S is called polyhedral metric if each point of S has a neighborhood isometric to a neighborhood of the vertex of a cone. In the paper the following two theorems are proved: Theorem 1: Every complete surface with polyhedral metric of negative curvature can be isometrically embedded into R3$R^3$ as a saddle surface. Theorem 2: Any polyhedral metric on a cylinder with a closed geodesic has a realization as a saddle surface in R3$R^3$ (a realization of a metric on a surface S in R3$R^3$ is a surface S′$S^{\prime }$ in R3$R^3$ and an isometric map f of S into R3$R^3$, which is not necessarily C0$C^0$, such that f(S)=S′)$f(S)=S^{\prime })$.

Translation from Sib. Mat. Zh. 28, No.4(164), 185-186 (Russian) (1987; Zbl 0623.52007).

Let M be a convex body in Euclidean n-space En$E^n$ and let k∈{1,...,n}$k\in \{1,...,n\}$. The symbol rk$r_k$ denotes the greatest radius of a k- dimensional ball contained in M. By Rk$R_k$ the author means the smallest radius of an (n+1−k)−$(n+1-k)-$ dimensional ball which contains an orthogonal projection of M onto an (n+1−k)−$(n+1-k)-$ dimensional plane. It is shown that Rk/rk≤k+1$R_k/r_k\le k+1$ for every convex body in En$E^n$ and that R2/r2≤2.15$R_2/r_2\le 2.15$ for every convex body in E3$E^3$.

The following metrical analog of the classical Helly theorem [Eh. Helly, Usp. Mat. Nauk 2, 80-81 (1936)] is proved by the authors: Theorem. Let D$\mathfrak{D}$ be a bounded family of compact convex sets in En$E^n$ such that μn(∩D)=0.${\mu }_n(\cap \mathfrak{D})=0.$ Then, for each ϵ>0$ϵ>0$ there exist n+1$n+1$ sets F1,...,Fn+1$F_1,...,F_{n+1}$ in D$\mathfrak{D}$ with μn(∩n+1i=1Fi)<ϵ;${\mu }_n({\cap }_{i=1}^{n+1}{F}_i)<ϵ;$ if, in addition, dim(∩D)=ℓ,$\dim (\cap \mathfrak{D})=\ell ,$ 0≤ℓ<n$0\le \ell <n$, then the initial number of n+1$n+1$ sets in D$\mathfrak{D}$ may be reduced to n−ℓ+1.$n-\ell +1.$
Some consequences of this result are also discussed.

[For the entire collection see Zbl 0603.00006.]
A preliminary result on facial structure of convex polytopes is discussed.