A note on exotic families of 4-manifolds
aa r X i v : . [ m a t h . G T ] J a n A NOTE ON EXOTIC FAMILIES OF 4-MANIFOLDS
TSUYOSHI KATO, HOKUTO KONNO, AND NOBUHIRO NAKAMURA
Abstract.
We present a pair of smooth fiber bundles over the circle witha common 4-dimensional fiber with the following properties: (1) their totalspaces are diffeomorphic to each other; (2) they are isomorphic to each otheras topological fiber bundles; (3) they are not isomorphic to each other assmooth fiber bundles.
In the previous paper [3] we showed that there exists a smooth manifold whichis realized as the total space of a non-smoothable topological fiber bundle (i.e. abundle with structure group the homeomorphism group which does not admit areduction to the diffeomorphism group) with a 4-dimensional fiber. The purposeof this short note is to point out that there exists a smooth manifold which canbe realized as the total spaces of two smooth fiber bundles with a common 4-dimensional fiber which are not mutually isomorphic as smooth fiber bundles, butisomorphic as topological fiber bundles:
Theorem 1.
There exist smooth fiber bundles E and E over S with a commonfiber X which is a smooth closed -manifold with the following properties: (1) the total spaces of E , E are diffeomorphic to each other; (2) E , E are isomorphic to each other as topological fiber bundles; (3) E , E are not isomorphic to each other as smooth fiber bundles.Proof. Ruberman [9] constructed a self-diffeomorphism f with the following prop-erties on a certain closed, oriented, simply-connected non-spin 4-manifold X , takento be the connected sum of copies of CP and copies of − CP : f is topologicallyisotopic to the identity of X , but not smoothly isotopic to the identity.Let E be the product bundle S × X over S and E be the mapping torus of f . By the properties of f , the conditions (2), (3) in the theorem are both satisfied,and it suffices to check the condition (1).Let { f t } t ∈ [0 , be a topological isotopy from the identity to f . We have a home-omorphism F : E → E induced from the map [0 , × X → [0 , × X definedby ( t, x ) ( t, f − t ( x )). Let i : X ֒ → E be the inclusion map onto the fiber overthe coset [0] ∈ S of 0 ∈ [0 , S be the natural smooth manifold structureon E , and S be the smooth manifold structure on E defined as the pull-backunder F of the natural smooth structure on the total space E . Define a topologicalmanifold M to be the underlying topological manifold of E , and denote by M j thesmooth manifold ( M, S j ) for j = 1 ,
2. Note that i is a smooth map with respect toboth of S and S . (However, an inclusion from X onto another fiber of E is notnecessarily a smooth map with respect to S , since f − t is not necessarily a smoothmap for t = 0 , X is a closed simply-connected 4-manifold, we have that i : X ֒ → M induces isomorphisms of second and fourth cohomology groups witharbitrary coefficient. In particular H ( M ; Z ) is torsion free. It follows from a theorem by ˇCadek and Vanˇzura [1, Theorem 1 (ii)] that theisomorphism class of an oriented vector bundle over M of rank 5 is determined bythe characteristic classes w , w , p . Here, to apply [1, Theorem 1 (ii)] to M , wehave used that H ( M ; Z ) is torsion free and that w ( M ) = 0, which follows fromthat X is non-spin. (See Remark in page 755 of [1].)We shall check the coincidence of these characteristic classes for T M and T M .Firstly, note that the normal bundle of the embedding i : X ֒ → M j is a trivialbundle even for j = 2, since real line bundles are classified by w and we have H ( X ; Z /
2) = 0. Thus we have that i ∗ w n ( M j ) = w n ( X ), and i ∗ are isomorphismsfor n = 2 , w n ( M ) = ( i ∗ ) − w n ( X ) = w n ( M ) for n = 2 ,
4. Similarly we have that p ( M ) = ( i ∗ ) − p ( X ) = p ( M ).Therefore it follows from [1, Theorem 1 (ii)] that T M and T M are isomorphic toeach other. This combined with the following lemma verifies the condition (1) inthe assertion of the theorem. (cid:3) Lemma 2.
Let E and E be smooth manifolds of the same dimension or .If there exists a homeomorphism F : E → E such that F ∗ T E is isomorphic to T E , then E and E are mutually diffeomorphic.Proof. Now we make use of some knowledge in high dimensional topology, mainlyknown as Kirby–Siebenmann theory [4]. As in the proof of Theorem 1, let M denote the underlying topological manifold of E , S denote the smooth manifoldstructure on E , S denote the smooth manifold structure on M defined as thepull-back under F of the smooth structure on E , and denote by M j the smoothmanifold ( M, S j ) for j = 1 ,
2. Let τ M : M → BT OP be the classifying map ofthe stable tangent microbundle. The two smooth structures M j on M give riseto lifts, which we denote by τ M j : M → BP L , of τ M along the natural map
BP L → BT OP . By the assumption that F ∗ T E is isomorphic to T E , we havethat T M and T M are mutually isomorphic, and thus we have that these lifts τ M j are homotopic to each other.Since dim M ≥
5, the homotopy classes of lifts of τ M are naturally in a bijectivecorrespondence with the concordance classes of PL structures on M . (See, forexample, [10, 1.7.2 Corollary].) Hence M and M are PL-concordant. By thefact that “concordance implies isotopy” in dimension ≥ M and M are PL isotopic, and in particular PL homeomorphic. By the fact that P L/O is6-connected and dim M ≤
6, smooth structures on the PL-manifold M are unique(see [10, 1.7.8 Remark]). Hence M is diffeomorphic to M . (cid:3) Remark . As far as H ( M ; Z ) is torsion free, one can avoid using the conditionthat X simply-connected in the proof of the coincidence of w , w , p for T M and T M . To to this, one can, instead, use the facts that the Stiefel–Whitneyclasses are homotopy invariant (see, for example, page 193 of [6]) and that therational Pontryagin classes are topological invariant, which is known as the Novikovtheorem [7]. Remark . It is classically known that, for an orientable closed smooth manifoldof dimension ≤
3, the inclusion from the diffeomorphism group into the homeo-morphism group is a weak homotopy equivalence. Hence a pair of fiber bundles E , E with such a lower dimensional fiber cannot satisfy the conditions (2), (3) inTheorem 1 at the same time. NOTE ON EXOTIC FAMILIES OF 4-MANIFOLDS 3
Acknowledgement.
Tsuyoshi Kato was supported by JSPS Grant-in-Aid for Sci-entific Research (B) No.17H02841 and JSPS Grant-in-Aid for Scientific Researchon Innovative Areas (Research in a proposed research area) No.17H06461. HokutoKonno was partially supported by JSPS KAKENHI Grant Numbers 16J05569,17H06461, 19K23412. Nobuhiro Nakamura was supported by JSPS Grant-in-Aidfor Scientific Research (C) No.19K03506.
References [1] Martin ˇCadek and Jiˇr´ı Vanˇzura,
On the classification of oriented vector bundles over -complexes , Czechoslovak Math. J. (1993), no. 4, 753–764. MR1258434[2] Morris W. Hirsch and Barry Mazur, Smoothings of piecewise linear manifolds , PrincetonUniversity Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. Annals of Math-ematics Studies, No. 80. MR0415630[3] T. Kato, H. Konno, and N. Nakamura,
Rigidity of the mod 2 families Seiberg-Witten invari-ants and topology of families of spin 4-manifolds , to appear in Compos. Math., available at arXiv:1906.02943 .[4] Robion C. Kirby and Laurence C. Siebenmann,
Foundational essays on topological mani-folds, smoothings, and triangulations , Princeton University Press, Princeton, N.J.; Univer-sity of Tokyo Press, Tokyo, 1977. With notes by John Milnor and Michael Atiyah; Annals ofMathematics Studies, No. 88. MR0645390[5] Ib Madsen and R. James Milgram,
The classifying spaces for surgery and cobordism of man-ifolds , Annals of Mathematics Studies, vol. 92, Princeton University Press, Princeton, N.J.;University of Tokyo Press, Tokyo, 1979. MR548575[6] J. P. May,
A concise course in algebraic topology , Chicago Lectures in Mathematics, Univer-sity of Chicago Press, Chicago, IL, 1999. MR1702278[7] S. P. Novikov,
On manifolds with free abelian fundamental group and their application , Izv.Akad. Nauk SSSR Ser. Mat. (1966), 207–246 (Russian). MR0196765[8] Frank Quinn, A controlled-topology proof of the product structure theorem , Geom. Dedicata (2010), 303–308, DOI 10.1007/s10711-009-9406-x. MR2721629[9] Daniel Ruberman,
An obstruction to smooth isotopy in dimension
4, Math. Res. Lett. (1998), no. 6, 743–758, DOI 10.4310/MRL.1998.v5.n6.a5. MR1671187 (2000c:57061)[10] Yuli Rudyak, Piecewise linear structures on topological manifolds , World Scientific PublishingCo. Pte. Ltd., Hackensack, NJ, 2016. MR3467983
Department of Mathematics, Faculty of Science, Kyoto University, KitashirakawaOiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan
Email address : [email protected] Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba,Meguro, Tokyo 153-8914, Japan, and, RIKEN iTHEMS, 2-1 Hirosawa, Wako, Saitama,351-0198, Japan
Email address : [email protected] Department of Mathematics, Osaka Medical College, 2-7 Daigaku-machi, TakatsukiCity, Osaka, 569-8686, Japan
Email address ::