In this note we determine all possible dominations between different products
of manifolds, when none of the factors of the codomain is dominated by
products. As a consequence, we determine the finiteness of every
product-associated functorial semi-norm on the fundamental classes of the
aforementioned products. These results give partial answers to questions of M.
Gromov.Comment: 7 pages; v2: small changes, to appear in Publicacions Matem\`atique

Neofytidis, Christoforos

English

arXiv

In this note we determine all possible dominations between different products of manifolds when none of the factors of the codomain is dominated byproducts. As a consequence, we determine the finiteness of every product-associated functorial semi-norm on the fundamental classes of the afore mentioned products. These results give partial answers to questions of M. Gromov

Diposit Digital de Documents de la UAB

Publ. Mat. 64 (2020), 653–660DOI: 10.5565/PUBLMAT6422011DOMINATION BETWEEN DIFFERENT PRODUCTSAND FINITENESS OF ASSOCIATED SEMI-NORMSChristoforos NeofytidisAbstract: In this note we determine all possible dominations between differentproducts of manifolds when none of the factors of the codomain is dominated byproducts. As a consequence, we determine the finiteness of every product-associatedfunctorial semi-norm on the fundamental classes of the aforementioned products.These results give partial answers to questions of M. Gromov.2010 Mathematics Subject Classification: 57N65, 55M25, 46B20.Key words: non-zero degree maps, semi-norms on products.1. Motivation and resultsA finite functorial semi-norm in degree k ∈ N singular homology is asemi-normν : Hk(X;R) −→ [0,∞)for every topological space X, where “functorial” means that the semi-norm ν is not increasing under induced homomorphisms f∗ : H∗(Y ) −→H∗(X) for all continuous maps f : Y −→ X.In [2, Chapter 5, Section G+] Gromov suggested (originally usingthe Euler characteristic of products of surfaces; see below) the followingconstruction of product-associated semi-norms on homology classes ofa topological space X: Let ν be a finite functorial semi-norm on thefundamental classes of products of closed oriented k-manifolds. For ahomology class α ∈ Hk`(X;Z), ` ∈ N, define(1) νk,`(α) := infd,M1×···×M`,fν([M1 × · · · ×M`])d,where the infimum is taken over all d = 1, 2, . . . , all products M1×· · ·×M` of closed oriented k-manifolds Mi, and all continuous maps f : M1×I am grateful to M. Gromov for stimulating questions and discussions. The hospitalityand support of the IHÉS, where part of this project was carried out, is also gratefullyacknowledged.654 C. Neofytidis· · · ×M` −→ X such that f∗([M1× · · ·×M`]) = d ·α. For ` = 1 we havea trivial product with only one factor.The idea of extending ν from the category of products of k-manifoldsto νk,`, i.e. to any k`-dimensional integral homology class and any topo-logical space X, stems from the following immediate property that everyfunctorial semi-norm satisfies:Lemma 1.1 (Mapping Lemma). Let k ≥ 1 and ν be a (finite) functo-rial semi-norm on the fundamental classes of products of closed orientedk-manifolds Mi. If f : M1 × · · · ×M` −→ M ′1 × · · · ×M ′` is a map ofdegree d, then ν(M1 × · · · ×M`) ≥ d · ν(M ′1 × · · · ×M ′`).Similarly to Thurston, who used the Euler characteristic of embeddedsurfaces in a 3-manifold M to define a norm in H2(M), Gromov’s originalexample in [2] is a norm in degree 2` homology, where ν is the absolutevalue of the Euler characteristic χ of (products of) surfaces. Namely, fora space X and α ∈ H2`(X;Z), ` ∈ N, the (product) Euler characteristicnorm is defined as(2) χ2,`(α) := infd,Σ1×···×Σ`,f|χ(Σ1 × · · · × Σ`)|d,where the infimum is taken over all d = 1, 2, . . . , all products Σ1×· · ·×Σ`of closed hyperbolic surfaces Σi, and all continuous maps f : Σ1 × · · · ×Σ` −→ X such that f∗([Σ1 × · · · × Σ`]) = d · α. Indeed, the Euler char-acteristic satisfies Lemma 1.1 for maps between (products of) surfaces;see the Mapping Lemmas in [2, Subsections 5.35–36].Gromov asked when the product Euler characteristic norm is finite,writing ([2, p. 301])“it is unclear which classes in H2`(X)come from (mapped) products of surfaces”.The obvious generalization of Gromov’s question is:Question 1.2. Let ν be a finite functorial semi-norm on the fundamen-tal classes of products of closed oriented k-manifolds. For which spaces Xand which homology classes α ∈ Hk`(X;Z), ` ∈ N, is νk,`(α) finite?Gromov predicted that the product Euler characteristic norm is infi-nite on many 2`-dimensional fundamental classes (` > 1), pointing outthe fundamental classes of irreducible locally symmetric spaces as po-tential candidates. That prediction has since been verified by Kotschickand Löh, who proved that irreducible locally symmetric spaces of non-compact type do not admit maps of non-zero degree from direct prod-ucts (whose factors are of any dimension, not necessarily surfaces); cf. [3,Corollary 4.2].Domination between Different Products 655The topic of realizing (co-)homology classes by direct products ofmanifolds is a special case of a classical problem of Steenrod [1, Prob-lem 25]. When the target homology class is the fundamental class of amanifold, we deal with maps of non-zero degree. We say that M domi-nates N , and write M ≥ N , if there is a continuous map f : M −→ N ofnon-zero degree, that is, f∗([M ]) = deg(f) · [N ] in homology or, equiva-lently, f∗(ωN ) = deg(f)·ωM in cohomology (as usual, ωM ∈ HdimM (M)denotes the cohomological fundamental class of M).The following question, posed to me by M. Gromov, is essential inorder to understand the finiteness of νk,` on the fundamental classesof arbitrary products, and has also independent interest at the level ofdomination between manifolds:Question 1.3. Let X1 × · · · × Xm be a Cartesian product of closedoriented manifolds Xi of positive dimensions. Which other non-trivialproducts dominate X1 × · · · ×Xm?In this paper we give a complete answer to Question 1.3 when noneof the factors Xi is dominated by products:Theorem 1.4. Suppose X1, . . . , Xm, Y1, . . . , Y` are closed oriented man-ifolds of positive dimensions such that X1, . . . , Xm are not dominated bynon-trivial direct products and dim(X1×· · ·×Xm) = dim(Y1×· · ·×Y`).Then Y1×· · ·×Y` ≥ X1×· · ·×Xm if and only if Yi ≥ Xai1 ×· · ·×Xaiξifor all i = 1, . . . , `, where ξi ≥ 1, aij ∈ {1, . . . ,m}, and aij 6= ai′j′ if(i, j) 6= (i′, j′).In particular, we obtain an answer to Question 1.2 for fundamentalclasses of products whose factors are not dominated by products:Corollary 1.5. Let X1, . . . , Xm be closed oriented manifolds of positivedimensions that are not dominated by non-trivial direct products anddim(X1×· · ·×Xm) = k`, for some k, ` ∈ N. The following are equivalent:(i) X1×· · ·×Xm is a product with ` factors of closed oriented k-man-ifolds.(ii) Every semi-norm νk,` is finite on [X1 × · · · ×Xm].(iii) There is a finite semi-norm νk,` on [X1 × · · · ×Xm].Note that if α = [M1 × · · · ×M`] in (1), then obviouslyνk,`(α) = ν([M1 × · · · ×M`]).Thus, the equivalent conditions (i)–(iii) in Corollary 1.5 are moreoverequivalent toνk,`([X1 × · · · ×Xm]) = ν([X1 × · · · ×Xm]),for every finite semi-norm ν.656 C. Neofytidis2. ProofsWe now prove Theorem 1.4 and Corollary 1.5. The proof of Theo-rem 1.4 uses Thom’s work [8] on the Steenrod problem about realizinghomology classes by closed manifolds [1, Problem 25]. Thom’s celebratedrealization theorem states that, given a topological space X and a homol-ogy class α ∈ Hk(X;Z), there is a closed oriented smooth k-dimensionalmanifoldM and a continuous map f : M −→ X such that f∗([M ]) = d·α,for some non-zero integer d. Or, equivalently, if one starts with a coho-mology class β ∈ Hk(X;Z), then f∗(β) = d · ωM .Proof of Theorem 1.4: The “if” direction is trivial and so we prove theconverse.Let f : Y1×· · ·×Y` −→ X1×· · ·×Xm be a map of non-zero degree, anddenote by pXi : X1 × · · · ×Xm −→ Xi the projection to the i-th factor.Then f∗(p∗Xi(ωXi)) is not trivial and, since the Xi are not dominated byproducts, Thom’s theorem ([8]) implies that f∗(p∗Xi(ωXi)) belongs toHdimXi(Y1;Q)⊕ · · · ⊕HdimXi(Yj ;Q)⊕ · · · ⊕HdimXi(Y`;Q).Indeed, suppose p∗Xi(ωXi) maps non-trivially under f∗ in someHk1(Y1;Q)⊗ · · · ⊗Hkj (Yj ;Q)⊗ · · · ⊗Hk`(Y`;Q),where 0 ≤ k1, . . . , k` < dimXi and k1 + · · · + k` = dimXi. Then, byThom’s theorem, there exist two closed oriented manifolds W1 and W2 ofpositive dimensions, with dimW1 + dimW2 = dimXi, and a continuousmap g : W1 ×W2 −→ Xi such that g∗(f∗(p∗Xi(ωXi))) = d · ωW1×W2 ∈HdimXi(W1 ×W2), for some non-zero integer d. Thus W1 ×W2 ≥ Xi,which contradicts our assumption that Xi is not dominated by products.Thus, we have(3) f∗(p∗Xi(ωXi)) =∑̀j=1(1× · · · × 1× αXij × 1× · · · × 1),where αXij ∈ HdimXi(Yj ;Q).A reduction: ` ≤ m. We first observe that (3) implies that ` can be atmost m, otherwise the number of factors in the codomain X1×· · ·×Xmof f would not suffice to givef∗(p∗X1(ωX1)) ∪ · · · ∪ f∗(p∗Xm(ωXm)) = deg(f) · ωY1×···×Y`= deg(f) · ωY1 × · · · × ωY` .(4)Domination between Different Products 657Thus we split the proof into the following cases:Case I: ` = m. In this case, (3) and (4) imply that for each Xi thereexists at least one Yj such thatdimXi = dimYj and f∗(p∗Xi(ωXi)) 6= 0 ∈ HdimYj (Yj ;Q).This means that Yj ≥ Xi through the composite mapYjιYj↪−−→ Y1 × · · · × Ymf−→ X1 × · · · ×XmpXi−−→ Xi,where ιYj : Yj ↪−−→ Y1 × · · · × Ym is the inclusion.The assumption that ` = m and (4) imply moreover that for each i 6=i′ there exist j and j′, j 6= j′, with Yj ≥ Xi and Yj′ ≥ Xi′ . Thus,after reordering the Yi if necessary, we conclude that Yi ≥ Xi for alli = 1, 2, . . . ,m.Case II: ` < m. In this case, (3) and (4) imply that for some Yi thereexist Xai1 , . . . , Xaiξi , ξi ≥ 2, among the X1, . . . , Xm, such thatξi∑j=1dimXaij = dimYiandf∗(p∗Xai1×···×Xaiξi(ωXai1×···×Xaiξi)) 6= 0 ∈ HdimYi(Yi;Q),where pXai1×···×Xaiξi: X1 × · · · ×Xm −→ Xai1 × · · · ×Xaiξi is the pro-jection. This means that Yi ≥ Xai1 × · · · ×Xaiξi through the compositemapYiιYi↪−−→ Y1×· · ·×Y`f−→ X1×· · ·×XmpXai1×···×Xaiξi−−−−−−−−−−→ Xai1×· · ·×Xaiξi ,where ιYi : Yj ↪−−→ Y1 × · · · × Y` is the inclusion.Now, by the naturality of the cup product, we obtainY1 × · · · × Yi−1 × Yi+1 × · · · × Y` ≥m∏q=1q/∈{ai1,...,aiξi}Xq,and so by the reduction established before we obtain ` − 1 ≤ m − ξi.If `−1 = m−ξi, then the result follows by Case I. If `−1 < m−ξi, thenwe repeat the argument as in Case II to find some i′ 6= i and some ξi′ ≥ 2such that Yi′ ≥ Xai′1 × · · · ×Xai′ξi′(where aij 6= ai′j′ for all 1 ≤ j ≤ ξi,1 ≤ j′ ≤ ξi′). We then have `− 2 ≤ m− ξi − ξi′ and we finish the proofby iterating the process.658 C. NeofytidisProof of Corollary 1.5: (i) ⇒ (ii) If X1 × · · · × Xm can be written asa product with ` factors of closed oriented k-manifolds Yi, then clearlyevery semi-norm νk,` is finite on [X1 × · · · ×Xm], becauseνk,`([X1 × · · · ×Xm]) = νk,`([Y1 × · · · × Y`]) = ν([Y1 × · · · × Y`])and ν([Y1 × · · · × Y`]) is finite by assumption.(ii) ⇒ (iii) This implication holds trivially.(iii) ⇒ (i) Suppose that some semi-norm νk,`([X1 × · · · ×Xm]) is finite.This means that there exist closed oriented k-manifolds Y1, . . . , Y` suchthatY1 × · · · × Y` ≥ X1 × · · · ×Xm.Then Theorem 1.4 implies that each Yi dominates a different (and pos-sibly containing only one factor) subproduct Xai1 × · · · ×Xaiξi ⊂ X1 ×· · · ×Xm. In particular, each Xai1 × · · · ×Xaiξi is a k-manifold, and soX1 × · · · ×Xm can be written as a product with factors those ` k-mani-folds:(Xa11×· · ·×Xa1ξ1 )×(Xa21×· · ·×Xa2ξ2 )×· · ·×(Xa`1×· · ·×Xa`ξ` ).Remark 2.1. The statements and proofs in this paper are at the levelof products of fundamental classes of manifolds. One can naturally gen-eralize Theorem 1.4 to the level of realizing arbitrary products of co-homology classes by other products of co-homology classes and, sub-sequently, obtain (non-)finiteness results of semi-norms on products ofmore general co-homology classes instead of fundamental classes of prod-ucts of manifolds.3. Two illustrative examplesThe key property in this paper is that none of the factors of thecodomain is dominated by direct products. There is a variety of exam-ples of manifolds that are not dominated by products, and techniques toidentify such manifolds were developed in the recent years [3, 4, 6, 7].Some large classes of examples are non-positively curved manifolds thatare not decomposable as products and certain circle bundles, includ-ing low-dimensional aspherical manifolds that possess certain Thurstongeometries. So, any combination of those manifolds can be used to con-struct direct products that fulfill Theorem 1.4 and Corollary 1.5.Example 3.1. Suppose X1, X2, and X3 are closed oriented manifolds ofdimensions dim(X1) = 3, dim(X2) = 6, and dim(X3) = 9. The possibleordered pairs (k, `) such that k` = dim(X1 ×X2 ×X3) = 18 are(1, 18), (2, 9), (3, 6), (6, 3), (9, 2), (18, 1).Domination between Different Products 659If the Xi are not dominated by products, then Corollary 1.5 applies:First, since there are three factors, then for any finite functorial semi-norm ν we obtainν1,18([X1×X2×X3]) = ν2,9([X1×X2×X3]) = ν3,6([X1×X2×X2]) =∞.Also X1 ×X2 ×X3 is not a product of three 6-manifolds, thus we haveν6,3([X1 ×X2 ×X3]) =∞.However, X1 × X2 × X3 is a product of the two 9-dimensional mani-folds X1 ×X2 and X3 and soν9,2([X1 ×X2 ×X3]) = ν([X1 ×X2 ×X3]) <∞.Finally, it is immediate by the definition in (1) thatν18,1([X1 ×X2 ×X3]) = ν([X1 ×X2 ×X3]) <∞.Example 3.2. Let X1, . . . , Xm, dim(Xi) ≥ 2, be closed oriented mani-folds that are not dominated by products. SupposeΣ1 × · · · × Σ` ≥ X1 × · · · ×Xm,where Σ1, . . . ,Σ` are closed oriented (hyperbolic) surfaces. By Theo-rem 1.4 (or by [5, Theorem 2.3]) we conclude that each Xi is a surface,and since the Xi are not dominated by products, we deduce that each Xiis a hyperbolic surface (and also m = `).Thus Corollary 1.5 implies that, if X1, . . . , Xm are closed orientedmanifolds that are not dominated by products, thenχ2,`([X1 × · · · ×Xm]) ={|χ(X1 × · · · ×Xm)| if each Xi is a surface,∞ otherwise.This answers Question 1.2 for the product Euler characteristic normon fundamental classes of products whose factors are not themselvesdominated by products.References[1] S. Eilenberg, On the problems of topology, Ann. of Math. (2) 50(2) (1949),247–260. DOI: 10.2307/1969448.[2] M. Gromov, “Metric Structures for Riemannian and Non-Riemannian Spaces”,Based on the 1981 French original, With appendices by M. Katz, P. Pansu, andS. Semmes, Translated from the French by S. M. Bates, Progress in Mathemat-ics 152, Birkhäuser Boston, Inc., Boston, MA, 1999.[3] D. Kotschick and C. Löh, Fundamental classes not representable by products,J. Lond. Math. Soc. (2) 79(3) (2009), 545–561. DOI: 10.1112/jlms/jdn089.[4] D. Kotschick and C. Löh, Groups not presentable by products, Groups Geom.Dyn. 7(1) (2013), 181–204. DOI: 10.4171/GGD/180.660 C. Neofytidis[5] D. Kotschick, C. Löh, and C. Neofytidis, On stability of non-dominationunder taking products, Proc. Amer. Math. Soc. 144(6) (2016), 2705–2710.DOI: 10.1090/proc/13043.[6] D. Kotschick and C. Neofytidis, On three-manifolds dominated by circle bun-dles, Math. Z. 274(1–2) (2013), 21–32. DOI: 10.1007/s00209-012-1055-3.[7] C. Neofytidis, Fundamental groups of aspherical manifolds and maps of non-zerodegree, Groups Geom. Dyn. 12(2) (2018), 637–677. DOI: 10.4171/GGD/451.[8] R. Thom, Quelques propriétés globales des variétés différentiables, Comment.Math. Helv. 28 (1954), 17–86.Department of Mathematics, Ohio State University, Columbus, OH 43210, USAE-mail address: neofytidis.1@osu.eduPrimera versió rebuda el 12 de juny de 2019,darrera versió rebuda el 6 de febrer de 2020.

Domination between different products and finiteness of associated semi-norms

RECERCAT

https://ddd.uab.cat/pub/pubmat/pubmat_a2020v64n2/pubmat_a2020v64n2p653.pdf

Domination between different products and finiteness of associated semi-norms

Abstract

Similar works

Full text

Available Versions

Diposit Digital de Documents de la UAB

RECERCAT