Homological theory of polytopes. Joseph Gubeladze San Francisco State University

Transcription

1 Homological theory of polytopes Joseph Gubeladze San Francisco State University

2 The objects of the category of polytopes, denoted Pol, are convex polytopes and the morphisms are the affine maps between them. Category of polytopes 2

3 The objects of the category of polytopes, denoted Pol, are convex polytopes and the morphisms are the affine maps between them. Let P V and Q W be polytopes in their ambient vector spaces. Category of polytopes 3

4 The objects of the category of polytopes, denoted Pol, are convex polytopes and the morphisms are the affine maps between them. Let P V and Q W be polytopes in their ambient vector spaces. Recall, a map f : P Q is an affine map if it respects barycentric coordinates or, equivalently, f is the restriction of a map of the form V W, x α(x) + w, where α : V W is a linear map, and w W. Category of polytopes 4

5 The objects of the category of polytopes, denoted Pol, are convex polytopes and the morphisms are the affine maps between them. Let P V and Q W be polytopes in their ambient vector spaces. Recall, a map f : P Q is an affine map if it respects barycentric coordinates or, equivalently, f is the restriction of a map of the form V W, x α(x) + w, where α : V W is a linear map, and w W. The set Hom(P, Q) of affine maps P Q is a polytope in a natural way in the ambient vector space of all affine maps V W. We call it the hom-polytope between P and Q. Category of polytopes 5

6 The objects of the category of polytopes, denoted Pol, are convex polytopes and the morphisms are the affine maps between them. Let P V and Q W be polytopes in their ambient vector spaces. Recall, a map f : P Q is an affine map if it respects barycentric coordinates or, equivalently, f is the restriction of a map of the form V W, x α(x) + w, where α : V W is a linear map, and w W. The set Hom(P, Q) of affine maps P Q is a polytope in a natural way in the ambient vector space of all affine maps V W. We call it the hom-polytope between P and Q. The facets of Hom(P, Q) are the maps P Q mapping a fixed vertex of P to a fixed facet of Q. One has dim Hom(P, Q) = (dim P + 1) dim Q. Category of polytopes 6

7 The objects of the category of polytopes, denoted Pol, are convex polytopes and the morphisms are the affine maps between them. Let P V and Q W be polytopes in their ambient vector spaces. Recall, a map f : P Q is an affine map if it respects barycentric coordinates or, equivalently, f is the restriction of a map of the form V W, x α(x) + w, where α : V W is a linear map, and w W. The set Hom(P, Q) of affine maps P Q is a polytope in a natural way in the ambient vector space of all affine maps V W. We call it the hom-polytope between P and Q. The facets of Hom(P, Q) are the maps P Q mapping a fixed vertex of P to a fixed facet of Q. One has dim Hom(P, Q) = (dim P + 1) dim Q. On the other extreme, the determination of the vertices of Hom(P, Q) is a real challenge. There are obvious vertex maps P Q, namely those mapping the whole polytope P to a vertex of Q. But, usually, Hom(P, Q) has many other vertices. Category of polytopes 7

8 A map P Q is a vertex of Hom(P, Q) if and only if it cannot be smoothly perturbed within Hom(P, Q), i.e., when f(p ) sits tightly in Q. Category of polytopes 8

9 A map P Q is a vertex of Hom(P, Q) if and only if it cannot be smoothly perturbed within Hom(P, Q), i.e., when f(p ) sits tightly in Q. Let P n denote the regular n -gon, n, n, n denote the unit n - dimensional simplex, cube, and cross-polytope, respectively. For a polytope P R d, containing 0 in the interior, we have the bipyramid (P ) = conv ( (P, 0), (0, 1), (0, 1) ) R d+1 and, if P is full-dimensional, the polar polytope P o. Category of polytopes 9

10 A map P Q is a vertex of Hom(P, Q) if and only if it cannot be smoothly perturbed within Hom(P, Q), i.e., when f(p ) sits tightly in Q. Let P n denote the regular n -gon, n, n, n denote the unit n - dimensional simplex, cube, and cross-polytope, respectively. For a polytope P R d, containing 0 in the interior, we have the bipyramid (P ) = conv ( (P, 0), (0, 1), (0, 1) ) R d+1 and, if P is full-dimensional, the polar polytope P o. The following series of explicit examples are found in [Bogart-Contois-G. 2013], which develops a systematic treatment of the hom-polytopes. Category of polytopes 10

11 A map P Q is a vertex of Hom(P, Q) if and only if it cannot be smoothly perturbed within Hom(P, Q), i.e., when f(p ) sits tightly in Q. Let P n denote the regular n -gon, n, n, n denote the unit n - dimensional simplex, cube, and cross-polytope, respectively. For a polytope P R d, containing 0 in the interior, we have the bipyramid (P ) = conv ( (P, 0), (0, 1), (0, 1) ) R d+1 and, if P is full-dimensional, the polar polytope P o. The following series of explicit examples are found in [Bogart-Contois-G. 2013], which develops a systematic treatment of the hom-polytopes. The vertex embeddings of P 4 P 7 up to symmetry: Category of polytopes 11

12 The vertex embeddings P 4 P 8 up to symmetry: Category of polytopes 12

13 The vertex embeddings P 4 P 8 up to symmetry: A vertex embedding of P 8 into 4-antiprism: Category of polytopes 13

14 Hom( n, P ) = P n+1 Hom(P, n ) = (P o ) n for P full-dimensional and centrally symmetric w.r.t. 0. In particular, Hom( m, n ) = ( m+1 ) n Hom( m, n ) = Hom( n 1, m+1 ) Category of polytopes 14

15 Hom( n, P ) = P n+1 Hom(P, n ) = (P o ) n for P full-dimensional and centrally symmetric w.r.t. 0. In particular, Hom( m, n ) = ( m+1 ) n Hom( m, n ) = Hom( n 1, m+1 ) As it turns out, the other cases of hom-polytopes between,, are considerably more challenging. They are considered in a separate paper. [G.-Love, 2015] For any m, n the polytope Hom( m, n ) has (n+1)(mn+ 1) vertices: the (n + 1) many maps from m to the vertices of n and the (n + 1)mn many compositions of the orthogonal projections m to the unit segment [0, 1] and the edge embeddings [0, 1] n. Category of polytopes 15

16 A visualization of Hom( 2, 3 ) : Category of polytopes 16

17 A visualization of Hom( 2, 3 ) : [G.-Love, 2015] (Informal statement) The vertices of Hom( m, n ) and Hom( m, n ) are also known: they are built out, in an explicit algorithmic way, of four simple types of vertex maps (including the vertex collapsings and edge projections, similar to the ones in the previous theorem), and the following distinguished vertex of Hom( 3, 3 ) : Category of polytopes 17

18 The distinguished vertex of Hom( 3, 3 ) : Category of polytopes 18

19 The distinguished vertex of Hom( 3, 3 ) : Remark. The vertices of Hom( m, n ) are not known. Category of polytopes 19

20 There is another natural construction in Pol, the tensor product of polytopes, which satisfies the natural conjunction: Category of polytopes 20

21 There is another natural construction in Pol, the tensor product of polytopes, which satisfies the natural conjunction: Hom(P, Hom(Q, R)) = Hom(P Q, R) Category of polytopes 21

22 There is another natural construction in Pol, the tensor product of polytopes, which satisfies the natural conjunction: Hom(P, Hom(Q, R)) = Hom(P Q, R) Up to isomorphism, P Q = conv ( (x y, x, y) x P, y Q ) ( V W ) V W, where P V and Q W are the ambient vector spaces. Category of polytopes 22

23 There is another natural construction in Pol, the tensor product of polytopes, which satisfies the natural conjunction: Hom(P, Hom(Q, R)) = Hom(P Q, R) Up to isomorphism, P Q = conv ( (x y, x, y) x P, y Q ) ( V W ) V W, where P V and Q W are the ambient vector spaces. As a result, similarly to the categories of sets or modules over a commutative ring, the category Pol is a closed symmetric monoidal category, enriched over itself. Category of polytopes 23

24 There is another natural construction in Pol, the tensor product of polytopes, which satisfies the natural conjunction: Hom(P, Hom(Q, R)) = Hom(P Q, R) Up to isomorphism, P Q = conv ( (x y, x, y) x P, y Q ) ( V W ) V W, where P V and Q W are the ambient vector spaces. As a result, similarly to the categories of sets or modules over a commutative ring, the category Pol is a closed symmetric monoidal category, enriched over itself. It is shown in [Bakuradze-Gamkrelidze-G., 2016] that the monoidal structure extends to the category of (finite) polytopal complexes, whose objects are obtained by gluing polytopes along affine isomorphisms of faces and morphisms are face-wise affine maps Category of polytopes 24

25 Examples of polytopal complexes, where (b) and (c) are built out of copies of 2 and, unlike the complex (a), they cannot be embedded into a Euclidean space by globally defined affine maps: ( a ) ( b ) ( c ) Category of polytopes 25

26 Examples of polytopal complexes, where (b) and (c) are built out of copies of 2 and, unlike the complex (a), they cannot be embedded into a Euclidean space by globally defined affine maps: ( a ) ( b ) ( c ) Next is a brief synopsis of [Gubeladze, 2016], which further explores categorial aspects of polytopes beyond Hom and in Pol. Category of polytopes 26

27 Modeling the approach in algebraic geometry and algebraic topology, one can ask (i) whether the representable functors of well-known polytopal objects have expected properties within the category of functors defined on Pol, and (ii) whether natural polytopal correspondences are representable functors, leading to new geometric objects. Category of polytopes 27

28 Modeling the approach in algebraic geometry and algebraic topology, one can ask (i) whether the representable functors of well-known polytopal objects have expected properties within the category of functors defined on Pol, and (ii) whether natural polytopal correspondences are representable functors, leading to new geometric objects. After the building blocks of the emergent new theory, i.e., the representable functors Hom(P, ) : Pol Pol and Hom(, Q) : Pol Pol op, have been tackled, one can turn to kernel-like objects. Category of polytopes 28

29 Modeling the approach in algebraic geometry and algebraic topology, one can ask (i) whether the representable functors of well-known polytopal objects have expected properties within the category of functors defined on Pol, and (ii) whether natural polytopal correspondences are representable functors, leading to new geometric objects. After the building blocks of the emergent new theory, i.e., the representable functors Hom(P, ) : Pol Pol and Hom(, Q) : Pol Pol op, have been tackled, one can turn to kernel-like objects. A natural candidate is the Billera-Sturmfels fiber polytope [Billera-Sturmfels, 1992]. For a map f : P Q in Pol its fiber is defined by Σf = 1 vol ( f(p ) ) { Γ } γdµ R m, Category of polytopes 29

30 Modeling the approach in algebraic geometry and algebraic topology, one can ask (i) whether the representable functors of well-known polytopal objects have expected properties within the category of functors defined on Pol, and (ii) whether natural polytopal correspondences are representable functors, leading to new geometric objects. After the building blocks of the emergent new theory, i.e., the representable functors Hom(P, ) : Pol Pol and Hom(, Q) : Pol Pol op, have been tackled, one can turn to kernel-like objects. A natural candidate is the Billera-Sturmfels fiber polytope [Billera-Sturmfels, 1992]. For a map f : P Q in Pol its fiber is defined by Σf = 1 vol ( f(p ) ) { Γ } γdµ R m, where: P R m, Q R n, vol( ) is the Euclidean (dim f(p )) -volume in the affine hull Aff(f(P )), Γ is the set of Borel measurable sections γ : f(p ) P of f, and µ -is the Borel measure on R n. Category of polytopes 30

31 This polytope, being a vast generalization the secondary polytope of Gelfand-Kapranov-Zelevinsky, has important applications in triangulation theory. It can be thought of as the average fiber of the map f Category of polytopes 31

32 This polytope, being a vast generalization the secondary polytope of Gelfand-Kapranov-Zelevinsky, has important applications in triangulation theory. It can be thought of as the average fiber of the map f A natural question is whether, for R and f : P Q in Pol, the polytopes Hom(R, Σf) and Σ Hom(R, f) are isomorphic, mimicking the functorial isomorphism Hom(, ker α) = ker Hom(, α) in abelian categories (e.g., vector spaces). Category of polytopes 32

33 This polytope, being a vast generalization the secondary polytope of Gelfand-Kapranov-Zelevinsky, has important applications in triangulation theory. It can be thought of as the average fiber of the map f A natural question is whether, for R and f : P Q in Pol, the polytopes Hom(R, Σf) and Σ Hom(R, f) are isomorphic, mimicking the functorial isomorphism Hom(, ker α) = ker Hom(, α) in abelian categories (e.g., vector spaces). However, the mentioned isomorphism fails already for low-dimensional polytopes. Namely, for Q = R a centrally symmetric polygon, P a slant truncated right 3-prism over the basis Q, and f the identity basis embedding of Q, one has Hom(R, Σf) = Σ Hom(R, f). Category of polytopes 33

34 This polytope, being a vast generalization the secondary polytope of Gelfand-Kapranov-Zelevinsky, has important applications in triangulation theory. It can be thought of as the average fiber of the map f A natural question is whether, for R and f : P Q in Pol, the polytopes Hom(R, Σf) and Σ Hom(R, f) are isomorphic, mimicking the functorial isomorphism Hom(, ker α) = ker Hom(, α) in abelian categories (e.g., vector spaces). However, the mentioned isomorphism fails already for low-dimensional polytopes. Namely, for Q = R a centrally symmetric polygon, P a slant truncated right 3-prism over the basis Q, and f the identity basis embedding of Q, one has Hom(R, Σf) = Σ Hom(R, f). [Gubeladze, 2016] develops an affine-compact version of the linear kernel, which leads to the correct version of the naive fiber equality. Category of polytopes 34

35 This polytope, being a vast generalization the secondary polytope of Gelfand-Kapranov-Zelevinsky, has important applications in triangulation theory. It can be thought of as the average fiber of the map f A natural question is whether, for R and f : P Q in Pol, the polytopes Hom(R, Σf) and Σ Hom(R, f) are isomorphic, mimicking the functorial isomorphism Hom(, ker α) = ker Hom(, α) in abelian categories (e.g., vector spaces). However, the mentioned isomorphism fails already for low-dimensional polytopes. Namely, for Q = R a centrally symmetric polygon, P a slant truncated right 3-prism over the basis Q, and f the identity basis embedding of Q, one has Hom(R, Σf) = Σ Hom(R, f). [Gubeladze, 2016] develops an affine-compact version of the linear kernel, which leads to the correct version of the naive fiber equality. By dualizing the diagrams, one derives the notion of the affine cokernel coker Aff f of a map f : P Q in Pol. But unlike the affine kernel, the latter construction (almost always) has a simple meaning: linear projection of Q along the affine hull Aff f(p ), erasing much of geometry. Category of polytopes 35

36 A more radical departure from the linear/affine setup leads to more interesting quotient objects. Category of polytopes 36

37 A more radical departure from the linear/affine setup leads to more interesting quotient objects. Motivated by the space of sandwiched simplices, introduced in [Mond-Smithvan Straten, 2003] for modeling stochastic factorizations in statistics, one introduces a pair of new functors: sandwiching and complementing. Category of polytopes 37

38 A more radical departure from the linear/affine setup leads to more interesting quotient objects. Motivated by the space of sandwiched simplices, introduced in [Mond-Smithvan Straten, 2003] for modeling stochastic factorizations in statistics, one introduces a pair of new functors: sandwiching and complementing. In the special case of two polytopes of same dimension Q P, the first functor associates to a polytope R the space of affine maps ϕ : R P, satisfying Q ϕ(r), and the second functor associates to R the space of affine maps ψ : R P, satisfying ψ(r) P \ int(q). Category of polytopes 38

39 A more radical departure from the linear/affine setup leads to more interesting quotient objects. Motivated by the space of sandwiched simplices, introduced in [Mond-Smithvan Straten, 2003] for modeling stochastic factorizations in statistics, one introduces a pair of new functors: sandwiching and complementing. In the special case of two polytopes of same dimension Q P, the first functor associates to a polytope R the space of affine maps ϕ : R P, satisfying Q ϕ(r), and the second functor associates to R the space of affine maps ψ : R P, satisfying ψ(r) P \ int(q). In the general case, informally, for an affine map f : Q P, the sandwiching functor makes f(q) a necessary target, like 0 in the category of vector spaces, and the complementing functor makes the interior of f(q) an impossible target, representing a fusion of the topological quotient P/f(Q) and affine maps. Category of polytopes 39

40 A more radical departure from the linear/affine setup leads to more interesting quotient objects. Motivated by the space of sandwiched simplices, introduced in [Mond-Smithvan Straten, 2003] for modeling stochastic factorizations in statistics, one introduces a pair of new functors: sandwiching and complementing. In the special case of two polytopes of same dimension Q P, the first functor associates to a polytope R the space of affine maps ϕ : R P, satisfying Q ϕ(r), and the second functor associates to R the space of affine maps ψ : R P, satisfying ψ(r) P \ int(q). In the general case, informally, for an affine map f : Q P, the sandwiching functor makes f(q) a necessary target, like 0 in the category of vector spaces, and the complementing functor makes the interior of f(q) an impossible target, representing a fusion of the topological quotient P/f(Q) and affine maps. This opens up possibilities for developing two nontrivial and complementary notions of quotient objects for Pol, at the expense of giving up the linear nature of the resulting objects: the new functors process polytopes and their affine maps into semi-algebraic sets and their affine maps. Category of polytopes 40

41 Important role in corelating the sandwiching and complementing functors is plaid by the class of affine-rigid polytopes. Category of polytopes 41

42 Important role in corelating the sandwiching and complementing functors is plaid by the class of affine-rigid polytopes. A polytope P is affine-rigid if every map from the boundary P to a vector space, affine on the facets F P, extends to an affine map on P. Category of polytopes 42

43 Important role in corelating the sandwiching and complementing functors is plaid by the class of affine-rigid polytopes. A polytope P is affine-rigid if every map from the boundary P to a vector space, affine on the facets F P, extends to an affine map on P. The isometric version of this property is the much studied rigidity property in metric geometry, going back to the Cauchy Theorem that all convex 3-polytopes are rigid, which was extended by Alexandrov to all higher dimensions in the 1950s. Category of polytopes 43

44 Important role in corelating the sandwiching and complementing functors is plaid by the class of affine-rigid polytopes. A polytope P is affine-rigid if every map from the boundary P to a vector space, affine on the facets F P, extends to an affine map on P. The isometric version of this property is the much studied rigidity property in metric geometry, going back to the Cauchy Theorem that all convex 3-polytopes are rigid, which was extended by Alexandrov to all higher dimensions in the 1950s. One can show that (i) all simple polytopes of arbitrary dimensions, except n -gons for n 4 are affine-rigid and, on the other extreme, (ii) no simplicial polytope of any dimension, except simplices, is affine-rigid. Category of polytopes 44

45 Important role in corelating the sandwiching and complementing functors is plaid by the class of affine-rigid polytopes. A polytope P is affine-rigid if every map from the boundary P to a vector space, affine on the facets F P, extends to an affine map on P. The isometric version of this property is the much studied rigidity property in metric geometry, going back to the Cauchy Theorem that all convex 3-polytopes are rigid, which was extended by Alexandrov to all higher dimensions in the 1950s. One can show that (i) all simple polytopes of arbitrary dimensions, except n -gons for n 4 are affine-rigid and, on the other extreme, (ii) no simplicial polytope of any dimension, except simplices, is affine-rigid. However, the class of affine-rigid polytopes is considerably larger than the class of simple polytopes. For instance, if the nonsimple vertices of a polytope P are very rarefied among the simple ones then P can have the property. Category of polytopes 45

46 Explicit examples are given by the anti-bipyramids: these 3-polytopes have two antipodal vertices, separated by the zig-zagging equators through the other vertices all simple: Category of polytopes 46

47 Explicit examples are given by the anti-bipyramids: these 3-polytopes have two antipodal vertices, separated by the zig-zagging equators through the other vertices all simple: A complete classification of the affine-rigid polytopes is not known. Category of polytopes 47

48 REFERENCES T. Bogart, M. Contois, and J. Gubeladze, Hom-polytopes, Math. Z., 273 (2013), J. Gubeladze and J. Love, Vertex maps between,, and, Geometriae Dedicata 176 (2015), M. Bakuradze, A. Gamkrelidze, and J. Gubeladze, Affine hom-complexes, Port. Math. 73 (2016), L. Billera and B. Sturmfels. Fiber polytopes. Ann. of Math. (2), 135 (1992), J. Gubeladze, Affine-compact functors, (preprint) D. Mond, J. Smith, and D. van Straten. Stochastic factorizations, sandwiched simplices and the topology of the space of explanations. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), Category of polytopes 48