EACAT7 at IISER, Mohali Homology of 4D universe for every 3-manifold

EACAT7 at IISER, Mohali Homology of 4D universe for every 3-manifold Akio Kawauchi Osaka City University Advanced Mathematical Institute December 2, 2017

1. Types I, II, full and punctured 4D universes Let M be the set of a closed connected orientable 3-manifold (or simply called a 3-manifold) M, and M 0 a compact punctured 3-manifold of M. Fact. For closed orientable 4-manifold X, M M such that M 0 is not embeddable in X, so that M is not embeddable in X. Cf. A. Kawauchi, The imbedding problem of 3-manifolds into 4-manifolds, Osaka J. Math. 25 (1988), 171-183.

Note: A difficulty on this non-embedding comes from the existence of a Samsara manifold on M. For m, n>0, let M m,n be the n-fold connected sum of the 0-surgery manifold of the m-fold connected sum of the trefoil knot. Then H * (M m,n ;Z)=H * (#ns 1 S 2 ;Z) and an orientation-preserving diffeomorphism h of M m,n such that h * =-1 on H 1 (M m,n ;Z). Let X be the mapping torus of h. Then M m,n is embedded in X with H * (X;Q)=H * (S 1 S 3 ;Q).

More generally, for M M, a Samsara manifold X on M, i.e., a compact oriented 4-manifold X with X=3-tori such that (0) M X with X-M connected, (1) H 2 (M;Q) H 2 (X;Q) is the 0-map, (2) Int X = 0: H 2 (X;Q) H 2 (X;Q) Q. X A.Kawauchi, Component-conservative invertibility of links and Samsara 4-manifolds on 3-manifolds, Asia Pacific J. Math., Vol. 1, No.2 (2014), 86-106. M Note 0: As noted on M m,n, closed Samsara manifolds X on lots of M M.

A 4D universe is a boundary-less orientable 4-manifold U with every 3-manifold embedded. If a 3-manifold M is a 3D universe and a smooth map t: M R is a time function, then a smooth embedding M M R sending x M to (x, t(x)), and M R is called the space-time of M. Since every embedded M in U admits a trivial normal line bundle M R in U, every 4D universe is considered as a classifying space for the spacetimes of all 3D universes M M.

Note on physical universe in low dimension Standard physical 4D universe model is Hyersphere World-Universe Model which is de Sitter space ds 4 =O(2,3)/O(1,3)= S 3 R (the spherical shell). In Brane cosmology, the physical 4D universe is a (3+1)-brane in a higherd bulk. Randall-Sundrum model is a physical 5D universe model showing that except Graviton, Elementary particles are localized on a (3+1)-brane or (3+1)- branes in the bulk which is anti-de Sitter 5D space AdS 5 =O(2,4)/O(1,4) (although Graviton is not yet confirmed).

Theorem 1.1. Let B 0 = S 3 R =S 3 R 0 B= S 3 R (-ε, ε) for ε>0. A 4D universe U is constructed from B 0 in B by collision recombinations of the sectional 3-spheres S 3 t, S 3 t (t t ) in B 0. S 3 S 3 or Collision recombinations

Idea of Proof: We show that 3-manifold M M is included in a collision recombination of two 3-spheres S 3 t, S 3 t (t t ) in B 0 by using Fact: M M is obtained by a Dehn surgery of S 3 along a 0-framed link of trivial components. //

A punctured 4D universe = a boundary-less orientable 4-manifold U with every punctured 3- manifold embedded. A punctured 4D universe is also considered as a classifying space for the spacetime M 0 R of every punctured 3D universe M 0.

Let X be a connected open orientable 4-manifold. two types of (smooth) embeddings M X. Definition. An embedding f: M X is of type 1 if X-f(M) is connected, and of type 2 if X-f(M) is disconnected. X X f(m) f(m) Type I Type II

Definition. Let U be a 4D universe. (1) U is of type 1 if M is type 1 embedded in U. (2) U is of type 2 if M is type 2 embedded in U. (3) U is full if U is of type 1 and type 2. Problem. Characterize the topological shapes of types 1, 2, full and punctured 4D universes. Remark. The 4D universe U constructed in Theorem 1.1 is taken as a full 4D universe.

Note 1: If type 1 embedding f:m U, then H 1 (U;Z) and H 3 (U;Z) have a direct summand Z, because x H 1 (U;Z) with Int U (x, f(m))=+1. Note 2: A full 4D universe is obtained from a 4D universe of type II by taking a connected sum with S 1 S 3. M M type I type II

Note 3: full full type I type II type I type II punctured punctured / / / / type I type II punctured punctured full full type I type II cf. [kawauchi 2015] A. Kawauchi, On 4-dimensional universe for every 3-dimensional manifold, Topology Appl., 196 (2015), 575-593.

Motivation of this talk. In [kawauchi 2015], several mutually independent non-negative integer invariants (later given) are introduced for all 4D universes or 4D punctured universes U to be + for at least one invariant of every U. However, I had failed to confirm that dim Q H 2 (U;Q)=+ for U. In this talk, this infinity is confirmed.

2. Main Result Let X be an orientable non-compact 4-manifold. Let β d (X) = Z-rank H d (X;Z) = dim Q H d (X;Q). Theorem 2.1. For X with β 2 (X) <+, M M such that M 0 is not embeddable in X. Corollary 2.2. For punctured 4D universe or 4D universe U, β 2 (U) =+. For the proof of Theorem 2.1, we apply the signature theorem for a non-compact 4-manifold in [kawauchi 2015] to a new family of 3-manifolds.

The d th null homology of a non-compact 4-manifold X: O d (X;Z)={x H d (X;Z) Int(x,H 4-d (X;Z))=0}. The d th non-degenerate homology of X: ^ H d(x;z) = H d (X;Z)/O d (X;Z) : Z-free ^ Let β ^ d(x) = Z-rank H d(x;z). ^ Assume that β 2(X) <+.

A homomorphism γ :H 1 (X;Z) Z is end-trivial if γ cl(x\a compact 4-submanifold)=0. Let (X,B ) be the infinite cyclic covering of (X,B) on γ, where B= X is a closed 3-manifold. For Γ=Q[t,t -1 ], consider the Γ-intersection form Let Int Γ : H 2 (X ;Q) H 2 (X ;Q) Γ. O 2 (X ;Q) Γ = {x H 2 (X ;Q) Int Γ (x, H 2 (X ;Q))=0}, ^ H 2(X ;Q) Γ =H 2 (X ;Q)/ O 2 (X ;Q) Γ (which is a free Γ-module of finite rank for an end-trivial γ ).

Let A(t) be a Γ-Hermitian matrix representing the Γ-intersection form Int Γ on ^ H 2(X ;Q) Γ. Let a, x (-1, 1). Define τ a±0 (X ) = lim x a±0 sign A(x+(1-x 2 ) 1/2 i). The signature invariants σ x (B ), x (-1, 1), of B is defined on the quadratic form ([kawauchi 1977]) b:tor Γ H 1 (B ;Q) Tor Γ H 1 (B ;Q) Q so that σ (a,1] (B ) = σ x (B ). a<x<1 [kawauchi 1977] A. Kawauchi, On quadratic forms of 3- manifolds, Invent. Math. 43 (1977), 177-198.

Signature Theorem ([kawauchi 2015]). τ a-0 (X )-sign X =σ [a,1] (B ), τ a+0 (X )-sign X =σ (a,1] (B ). Corollary ([kawauchi 2015]). For a (-1,1), σ [a,1] (B ) - κ 1 (B ) 2 ^ β 2 (X;Z), where κ 1 (B ) = dim Q Ker(t-1:H 1 (B ;Q) H 1 (B ;Q)).

Proof of Theorem 2.1. For X with β 2 (X) =d<+, we show that M M such that M 0 is not embeddable in X. Suppose M 0 is in X for an M M with β 1 (M)=n. The 2-sphere S 2 = M 0 is a 2-knot in X. Let X M be the 4-manifold obtained from X by replacing N(K) =S 2 D 2 by D 3 S 1. Then β 2 (X M ) = β 2 (X) = d and M is embedded in X M by a type 1 embedding.

A homological 3-torus is M=M(k 1,k 2,k 3 ) M obtained from the 3-torus T 3 and 3 knots k 1,k 2,k 3 in S 3 by replacing the 3 solid torus axes with the knot exteriors E(k 1 ), E(k 2 ), E(k 3 ). Then a Z-basis u 1,u 2,u 3 H 1 (M;Z) with u 1 u 2 u 3 H 3 (M;Z)=Z a generator. For m>0, let T m be a collection of the m connected sum M =# i=1 M(k i1,k i2,k i3 ) M for knots k i1,k i2,k i3 (i=1,2,,m) in S 3. Then we show that M T m non-embeddable in X M by a type 1 embedding.

For the 4-manifold X obtained from X M by splitting along M, let i : M M (-1) X and i: M M 1 X be the natural maps. For i *, i * :H 2 (M;Q) H 2 (X ;Q), let C= Im i * Im i *. We show: Lemma 2.2. m 0 >0 such that 3m 0 >d and every M T m with m>m 0 does not satisfy one of (1)-(3). (1) i * and i * are injective, (2) C*= i * -1 (C)=i * -1 (C) 0, (3) C* has a Q-basis a 1,a 2,,a s with i * (a i )=±i * (a i ) for i.

First, assuming Lemma 2.2, we show that M T m non-embeddable in X M by a type 1 embedding. Let B= M 1 M (-1)= X. A homomorphism γ : H 1 (B;Z) Z is Q-asymmetric if there are no Z-linearly independent elements x 1,x 2,,x n H 1 (M;Z) such that γ (x i )=±α * (x i ) for i, where α is the reflection in B.

By Lemma 2.2, one of the following (1), (2) and (3) occurs: (1) i * or i * is not injective. (2) i * and i * are injective, and C*=0 or i * -1 (C) i * -1 (C). (3) i * and i * are injective, (i * ) -1 (C)=(i * ) -1 (C)=C* 0 and there is no Q-basis a 1,a 2,,a s of C* such that i * (a i ) =± i * (a i ) for i.

If i * and i * are injective and C*=0, then H 2 (M;Q) H 2 (X ;Q) is injective. Since β 1 (M)=n=3m>d=β 2 (X)=d, we have C* 0 and hence i -1 * (C) i -1 * (C). Then in either case, we see that an end-trivial homomorphism γ :H 1 (X ;Z) Z such that γ B: H 1 (B;Z) Z is Q-asymmetric.

^ Since β 2(X) d and ^ ^ β 2(X ) β 2(X), Signature Theorem implies: σ [a,1] (B ) - κ 1 (B ) 2d for all a. Choose knots k i1,k i2,k i3 (i=1,2,,m) so that σ(k 11 ) >2d+2m, σ(k ij ) > (i,j)>(i,j ) σ(k i j ) +2d+2m (i, i = 2,3,,3m; j, j = 1,2,3). (Note: Suitable connected sums of copies of a trefoil knot can be taken as these knots.)

Then we see that κ 1 (B ) 2m and σ [a,1] (B ) >2d+2m for some a, so that M T m non-embeddable in X M by a type 1 embedding. This completes the proof of Theorem 2.1 except for the proof of Lemma 2.2.

Proof of Lemma 2.2. Let X be the infinite cyclic cover of X associated with (X ; M (-1), M 1), and n=3m. (*) Suppose that i * and i * are injective, (i * ) -1 (C)=(i * ) -1 (C)=C* 0 and there is a Q-basis a 1,a 2,,a s of C* such that i * (a i ) =± i * (a i ) for i. Then H 2 (X ;Q) = Γ d + (Γ/(t+1)) c(+) + (Γ/(t-1)) c(-) where dim Q C = c(+)+c(-) n, n-(c(+)+c(-)) d, d +c(-) d, so that n-c(+) d.

Let Y be a compact 4-manifold such that M Y X and the Γ-torsion part TH 2 (Y ;Q) of the homology H 2 (Y ;Q) has TH 2 (Y ;Q)=TH 2 (X ;Q) = (Γ/(t+1)) c(+) + (Γ/(t-1)) c(-). By the duality in [kawauchi 1977], we have TH 1 (Y, Y ;Q) =(Γ/(t+1)) c(+) + (Γ/(t-1)) c(-).

Let M T m. Let H * (Y, Y ;Q)= TH * (Y, Y ;Q) + F * (Y, Y ;Q) be any splitting into the Γ-torsion part and Γ-free part. Let H * (Y, Y ;Q) = T * (Y, Y ;Q) + F * (Y, Y ;Q) be the Q-dual splitting. Let T 1 (Y, Y ;Q) t+1 be the (t+1)-component of T 1 (Y, Y ;Q).

Consider the cup product T 1 (Y, Y ;Q) t+1 T 1 (Y, Y ;Q) t+1 H 2 (Y, Y ;Q) k* k* k* H 1 (M;Q) H 1 (M;Q) H 2 (M;Q). Consider the Q-subspace Ω = {{k*(u v) H 2 (M;Q) u, v T 1 (Y, Y ;Q) t+1 }} Q. Let ω = dim Q Ω. Lemma 2.3. ω d and lim m + ω = +. Since d is fixed, we have a contradiction. This implies Lemma 2.2.

(Proof of Lemma 2.3) Let j*: H 2 (Y, Y ;Q) H 2 (Y ;Q) and k *: T 2 (Y ;Q) H 2 (M;Q). By a transfer argument of: A. Kawauchi, On the signature invariants of infinite cyclic coverings of closed odd dimensional manifolds, Algebraic and Topological Theories, Kinokuniya (1985), 52-85. k *:T 2 (Y ;Q) H 2 (M;Q) is injective. Since k*(u v)=k * j*(u v) k *T 2 (Y ;Q) t-1, we have Ω k *T 2 (Y ;Q) t+1 =0. Hence Ω H 2 (M;Q)/k *T 2 (Y ;Q) t+1 is injective.

Since T 2 (Y ;Q) t+1 = (Γ/(t+1)) c(+), we have ω dim Q H 2 (M;Q)/k *T 2 (Y ;Q) t+1=3m-c(+) d. Further, since dim Q T 1 (Y, Y ;Q) t+1 = c(+) and k*: T 1 (Y, Y ;Q) t+1 H 1 (M;Q) is also a monomorphism and 3m-d c(+), a Q-basis of H 1 (M;Q) whose only at most d elements are not in k*t 1 (Y, Y ;Q) t+1. Since M T m, we have lim m + ω = + by a Q-basis change argument of the Q-basis, showing Lemma 2.3. // This completes the proof of Theorem 2.1. //

3. Earlier topological indexes on 4D universes The following invariants are in [kawauchi 2015] : Given a punctured 4D universe U and M 0 M 0, let δ(m 0 U) =min Z-rank(im[ι * :H 2 (M 0 ;Z) H 2 (U;Z)]), ρ(m 0 U) = min Z 2 - rank (im[ι * :H 2 (M 0 ;Z) H 2 (U;Z)] (2) ) for all embeddings ι: M 0 U. Define δ 0 (U) =sup{δ(m 0 U) M 0 M 0 }, ρ 0 (U) = sup{ρ(m 0 U) M 0 M 0 }.

Similarly, given a 4D universe U of type 1 and M M, let δ (M 1 U) =min Z-rank(im[ι * :H 2 (M;Z) H 2 (U;Z)]), ρ(m 1 U) =min Z 2 -rank(im[ι * :H 2 (M;Z) H 2 (U;Z)] (2) ) for all type 1 embeddings ι: M U. Let δ 1 (U) =sup{δ(m 1 U) M M}, ρ 1 (U) =sup{ρ(m 1 U) M M}.

Given a 4D universe U of type 2 and M M, let β 2 (M 2 U)=min Z-rank(im[ι * :H 2 (M;Z) H 2 (U;Z)]), β (2) 2 (M 2 U)=min Z 2 -rank (im[ι * :H 2 (M;Z) H 2 (U;Z)] (2) ) for all type 2 embeddings ι: M U. Let δ 2 (U) =sup{β 2 (M 2 U) M M}, ρ 2 (U) =sup{β (2) 2 (M 2 U) M M}.

Given a 4D universe or a full 4D universe U and M M, let δ(m U) =min Z-rank(im[ι * :H 2 (M;Z) H 2 (U;Z)]), ρ(m U)=min Z-rank(im[ι * :H 2 (M;Z) H 2 (U;Z)] (2) ) for all embeddings ι: M U. Let δ(u) =sup{δ(m U) M M}, ρ(u) =sup{ρ(m U) M M}.

4. How is any 4D universe infinite? Combining Theorem 2.1 with [kawauchi 2015], we obtain: Infinity Theorem. (1) If U is a punctured 4D universe, then β 2 (U) =+ and U has one of the cases (i)-(iii). (i) ^ β 2(U) =+. (ii) δ 0 (U) =+. (iii) ρ 0 (U) =+. Further, for every case in (i)-(iii), a punctured spin 4D universe U with the other indexes taken 0.

(2) If U is a 4D universe of type 1, then β 2 (U) =+ and U is in one of the cases (i)-(iii). (i) ^ 2(U) =+ and ^ β β 1(U) >0. (ii) δ 1 (U) =+ and ^ β 1(U) >0. (iii) ρ 1 (U) =+ and ^ β 1(U)=+. Further, for every case in (i)-(iii), a type 1 spin 4D universe U with the other topological indexes on ^ β 2(U), δ 1 (U) and ρ 1 (U) taken 0.

(3) If U is a 4D universe of type 2, then β 2 (U) =+ and U is in the case (i) or (ii). (i) ^ β 2(U) =+. (ii) δ 2 (U) =+. Further, for every case in (i)-(ii), a type 2 spin 4D universe U with the other topological index taken 0.

(4) If U is a 4D universe, then β 2 (U) =+ and U is in one of the cases (i)-(iii). (i) ^ β 2(U) =+. (ii) δ(u) =+. ^ (iii) ρ(u) =+ and β 1(U)=+. Further, for every case in (i)-(iii), a spin 4D universe U with the other topological indexes on ^ β 2(U), δ(u) and ρ(u) taken 0.

(5) If U is a full 4D universe, then β 2 (U) =+ and U is in one of the cases (i)-(ii). (i) ^ β 2(U) =+ and ^ β 1(U) >0. (ii) δ(u) =+ and ^ β 1(U) >0. Further, for every case in (i)-(ii), a full spin 4D universe U with the other topological index on ^ β 2(U) and δ(u) taken 0.

Final note: The infinity in every case of a 4D universe comes from the existence of the trefoil knot. Thank you for your attention.