Scissors Congruence in Mixed Dimensions

Transcription

1 Scissors Congruence in Mixed Dimensions Tom Goodwillie Brown University Manifolds, K-Theory, and Related Topics Dubrovnik June, 2014

2 Plan of the talk I have been exploring the consequences of a definition. Outline: Rings of polytopes Multiplicative invariants New invariants from old Speculations and further directions

3 The ring of Euclidean polytopes Consider bounded convex polytopes P R n, dim P n. Definition E n is the abelian group with a generator (P) n for every such P, and with generating relations (P Q) n = (P) n + (Q) n (P Q) n ( ) n = 0 (gp) n = (P) n if g is an isometry. Inclusion map E n 1 E n (P) n 1 (P) n. The cokernel is the n th scissors congruence group. Maybe these maps are injective. Call the direct limit E. Maybe E is the direct sum of all the scissors groups. Even so, the graded group E is a good thing.

4 Multiplication The graded group E is an associative, commutative, unital graded ring, with multiplication E p E q E p+q given by (P) p (Q) q = (P Q) p+q The ring E 0 is Z, generated by 1 = ( ) 0 E 0. The inclusion map E n 1 E n is multiplication by ( ) 1 E 1. Call this element p. Then there is the local analogue....

5 The local version An analogous construction: Instead of bounded Euclidean polytopes, use germs of Euclidean polytopes at a point. Another way to say this is: use conical polytopes. Or spherical polytopes. Definition Let L n be the abelian group with a generator (P) n for every such germ of dimension at most n, and with generating relations as before. Again these form a graded ring. We have L 0 = Z. When σ is a simplex in a Euclidean polytope P, then the cone on the link of σ in P determines such a germ, which we call ν P σ, the normal cone of σ in P. The inclusion (P) n 1 (P) n is multiplication by the element t = ( ) 1 L 1. Its cokernel is the (n 1) st spherical scissors congruence group.

6 Non-convex polytopes (P) n E n can be defined for more general sets P. The polytope P need not be convex, and need not be in R n. It can even be an abstract compact PL space (finite simplicial complex) with Euclidean structure. There is a unique way to do this that is compatible with the defining relations. We can go even further: to sets of the form P A where P is a Euclidean polytope and A P is a PL subspace. For example, we can write (int M) n = (M M) n = (M) n ( M) n if M is a compact PL manifold (of dimension n, with Euclidean structure) with boundary. Similar remarks apply to the local version.

7 Multiplicative invariants We study multiplicative invariants of polytopes, meaning ring maps Two obvious examples: F : E k. The Euler characteristic gives a ring homomorphism χ : E Z. Note that χ treats (P) n and (P) n+1 the same; we have χ(p) = 1. Call such invariants absolute. n-dimensional volume gives a group homomorphism V n : E n R, and these together give a ring homomorphism V : E R. This invariant is not absolute. In fact V (p) = 0.

8 Multiplicative invariants There are also multiplicative invariants L k in the local case. We begin with the following three: The spherical volume U : L R. This takes a cone in R n to the volume of its intersection with S n 1, normalized so that the volume of the whole sphere is one. It is well defined on the spherical scissors congruence ring L/tL. The reduced Euler invariant e : L Z, which takes the germ of P at 0 to χ(p, P 0) = 1 χ(p 0). This takes a d-dimensional vector space to ( 1) d and takes every other kind of convex cone to zero. It is an absolute invariant; we have e(t) = 1. The seemingly innocuous counting invariant 1 : L Z, which takes every polytope germ to 1.

9 The dual of a cone Definition For a convex cone P in R n with apex at the origin, its dual DP consists of all vectors v R n such that for every w P the inner product satisfies v, w 0. This respects products. So another multiplicative invariant, W : L R, the dual volume, is given by (P) n U n (DP) n for a convex cone P R n. This invariant is absolute. In fact there is an involution D : L L of the graded ring, given by (P) n (DP) n for convex P. So we can describe W as the composition U D of two ring maps L L R.

10 Making more invariants Our main theme is a way of making new invariants from old. The idea of this can be seen in the Dehn invariant for three-dimensional polytopes: P θ e V 1 (e) (R/Z) Z R. e Here V 1 (e) is the length of the edge, and θ e is the dihedral angle at that edge.

11 Extended volume The same idea can be seen in the following construction: The volume V 1 : E 1 R can be extended to a map V (1) : E 2 R. For bounded convex P R 2, define V (1, P) to be We have one half of the perimeter of P, if P is a planar polygon the length of P, if P is a line segment 0, if P is a point or empty V (1, P Q) = V (1, P) + V (1, P) V (1, P Q). Extend V (1) further, to E 3 : If P is three-dimensional then define V (1, P) as a sum over edges of P: ( 1 2 θ e) V 1 (e). e

12 Extended volume General rule for extending V n : E n R to a homomorphism V (n) : E R. Let P be convex, of any dimension. Define V (n, P) = F W (ν P F )V n (F ), a sum over n-dimensional faces of P. This is compatible with the defining relations and so gives an additive homomorphism V (n) : E R whose restriction to E n is n-dimensional volume.

13 Extending 0-dimensional volume V (0) is the Euler characteristic.

14 The Dehn invariant These R-valued invariants can be refined to (R Q R)-valued invariants by writing W (ν P F ) V n (F ) F with tensor product over Z rather than multiplication of real numbers. Thus for example when n = 1 we obtain a map E 3 R R that knows all about both the Dehn invariant and the extended length function E 3 /pe 2 R (R/Q) V (1) : E 3 R.

15 Low dimensional elements We have E 1 = Z R with the first summand detected by χ and the second summand detected by V 1. The inclusion E 1 E 2 splits and E 2 = E1 R = Z R R. E 2 has a point part, a length part, and an area part. The fact that the new part, E 2 /pe 1, is isomorphic to R is the elementary fact that two polygons having the same area are scissors congruent.

16 Low-dimensional elements The quotient E 3 /pe 2 is not simply R; scissors congruence in dimension three is not just a question of volume. Besides V 3 : E 3 R there is the Dehn invariant. In fact, E 3 = Z R?? where?? is a subgroup of R Q R.

17 Low-dimensional elements L 1 = Z Z, with a basis {t, d}, where d is given by the germ of a line segment at an endpoint. If s is given by the germ of a line segment at an interior point, then s = 2d t. Let d be d t, the germ of an open line segment at its absent boundary point.

18 Low-dimensional elements L 2 = L1 R = Z Z R. The quadratic monomials t 2, td, and d 2 are linearly independent. The first two of these span a summand tl 1 = Z Z. For an angle θ, let σθ 2 be the germ of a 2-simplex at a corner point, and define a(x) L 2 by a(θ) = (σ 2 θ ) 2 td. These pure angle elements satisfy a(θ + φ) = a(θ) + a(φ) and establish an isomorphism between R and the quotient L 2 /tl 1. In the subgroup a(r) L 2 the element d 2 td is a( 1 4 ). The element s 2 t 2 (germ of a plane minus germ of a point) is a(1).

19 Interior Operator I : E n E n. Takes a convex polytope P to its interior, up to a sign depending on the dimension of the polytope: ( 1) m I(P) n = (int P) n = (P) n ( P) n where m = dim(p). In addition to being well defined (satisfying the additivity relation), this has the following properties: I : E E is a map of graded rings. I is an involution The equation above is valid whenever P is an m-dimensional PL manifold (with Euclidean structure). For a triangulation of P we have I(P) n = Σ σ ( 1) σ (σ) n where σ runs through simplices of the triangulation and σ is the dimension of σ. Note that on the scissors group E n /pe n 1 the map I is ( 1) n.

20 Boundary Let ɛ be the grading involution of E (or any graded ring). It acts on E n by ( 1) n. Boundary operator δ : E n E n 1, given by δ(p) n = ( P) n 1 if the convex polytope (or manifold) P has dimension n 2k n, and by δ(p) n = (DP) n 1 if P has dimension n 1 2k < n. Here DP is the double. The operator δ satisfies pδ(x) = x ɛi(x) δ(px) = x + ɛi(x) δ(xy) = δ(x)y + I (x)δ(y) δ δ = 0 δ(p) = 2

21 Local interior and boundary The local version has its own involution I : L L and boundary operator δ : L n L n 1. On L 1 and L 2 they are given by I(t) = t, I(s) = s, I(d) = d δ(t) = 2, δ(s) = 0, δ(d) = 1. I(a(θ)) = a(θ), δ(a(θ)) = 0.

22 Duality and interior The duality involution of L is related to I by D I = ɛ I D.

23 Duality and suspension In L 1 we have Dt = s Ds = t Thus, for example, D takes t(p) n = (P) n+1 to s(dp) n = (R DP) n+1. In other words, the two obvious maps L n L n+1 ( inclusion and suspension ) are interchanged by D. Recall also that It = t Is = s

24 Eigenspaces of I After inverting 2 we can of course split E n as E + n E n where I acts like +1 and 1 on the two parts. Geometrically this corresponds to writing a manifold as the average of its double and its boundary; E + and E contain respectively the closed manifolds of even and odd dimension. Lkewise L n = L + n L n. When D is even, I preserves L + n interchanges L + n and L n. and L n. When n is odd, D

25 Scaling There is a trivial but useful way of making new invariants from old ones: if F : E k and c k, define c F : E k, the scaling of F by c: ( c F ) n (P) = c n F n (P). For example, think of converting feet to inches, square feet to square inches, cubic feet to cubic inches....

26 The groupoid Here is our main construction. Suppose that F : L k and G : E k are multiplicative invariants (local and global, respectively), and that F (s) = G(p). Definition The Dehn product F G : E k is defined by (F G) n (P) = σ F n σ (ν σ P) G σ (int σ), where σ runs through all simplices of a triangulation of P. The right hand side is in fact independent of the triangulation, and this is where we need the assumption that F (s) = G(p). We have (F G)(p) = F (t). We may say that G is located at G(p), and that F takes invariants located at F (s) to invariants located at F (t).

27 The groupoid For example, W takes invariants located at 0 to invariants located at 1 (absolute invariants). Recall that V is located at 0. Thus x V is also located at 0. The invariant W x V is defined, and its value at P is W (ν P σ) V σ (σ)x σ. σ P The coefficient of x n is our old friend V (n), the extended n-dimensional volume: W (ν P σ) V n (σ) = V (n, P). Thus σ =n (W x V )(P) = V (0, P) + V (1, P)x + + V (n, P)x n +... The additive invariants V (0), V (1),... are being packaged as a single multiplicative invariant. The multiplicativity means that V (n, P Q) = V (p, P) V (q, Q). p+q=n

28 The groupoid We defined the global invariant F G when F was local and G was global, assuming F (s) = G(p). Definition The Dehn product F G of two local invariants is defined by the same formula. It is defined whenever F (s) = G(t). For each ring k this partially defined multiplication law on the set of ring maps L k gives us a category, in fact a groupoid.

29 The groupoid Definition Let k be a ring. The groupoid G(k) has the elements of k as objects. A morphism a b is a multiplicative local invariant F such that F (s) = a and F (t) = b. Composition is given by Dehn product. The identity morphism of the object a is the scaling a 1 of the counting invariant. The inverse of a morphism F is F D, the composition with the duality involution. The groupoid depends functorially on the ring k. In fact, G is an affine groupoid scheme whose object space is the affine line. The scaling of invariants by elements of k extends to an action of the multiplicative monoid of k. This reflects the fact that G is made of graded rings and graded ring maps.

30 The groupoid 1 is a Z-valued morphism from 1 to 1 (in fact, the identity). e is a Z-valued morphism from 1 to 1. U is an R-valued morphism from 1 to 0. W, the inverse of U, is an R-valued morphism from 0 to 1. More generally, a 1 is the identity morphism from a to a. a e is a morphism from a to a. Its inverse is a e. If x R then x U is an R-valued morphism from x to 0. Or, letting x denote an indeterminate, x U is an R[x]-valued morphism from x to 0. W x U is an R[x]-valued morphism from x to 1.

31 The groupoid Thus (W x U)(P) = U(0, P) + U(1, P)x + + U(n, P)x n +... The additive invariants U(0), U(1),... are the coefficients of a single polynomial-valued multiplicative invariant. The multiplicativity means that U(n, P Q) = U(p, P) U(q, Q). p+q=n U(n) is an extension of U n from L n to L. U(0) = W. If P is a convex cone in R n then U(k, DP) = U(n k, P).

32 The groupoid Write x, y = x W y U. This invariant, which takes (P) n L n to U(0, P)x n + U(1, P)x n 1 y + + U(n 1, P)xy n 1 + U(n, P)y n, plays a crucial role. It satisfies x, y y, z = x, z x, x = x 1 c x, y = cx, cy. This family of invariants in effect reduces the groupoid G(k) to a group whenever the ring k is an R-algebra. But if the same ring is an R-algebra in more than one way then we can make more invariants.

33 The groupoid Standard ring maps α j : R R k, 1 j k. The product α 1 x 0, x 1 α 2 x 1, x 2 α k x k 1, x k gives a multiplicative invariant with values in R k [x 0,..., x k ]. This is nontrivial on L n as soon as n is roughly 2k.

34 The role of the Euler invariant e 1, 1 = e, and more generally x, x = x e. The invariant e is closely related to the interior operator I: For any multiplicative invariant F (local or global), the composition with I satisfies F ɛi = e F. In combination with the equation I D = ɛ D I this gives e F = F I = F e. So in some sense e (or rather the scalings of e taken all together) is central in the groupoid, and an involution.

35 Further ideas Questions: Are the inclusion maps injections? Does E have 2-torsion? This is related to the previous question. Is E generated by closed manifolds? True up to a factor of 2. Is all of E detected by the R k [x 0,..., x k ]-valued invariants described above?

36 Further ideas It is tempting to divide by the central involution e of the groupoid. This leads to something like a smaller ring than L consisting of the part generated by manifolds without boundary and having even dimension and codimension. There should be an intrinsic description of this. Up to 2-torsion it is the part of L fixed by both I and DID. The ring L is more or less obtained from it by adjoining roots of two quadratic equations: t and d.

37 Further ideas Applications to Riemannian manifolds? Combinatorial characteristic classes?