Homotopy theoretic properties of open books

We study the homotopy groups of open books in terms of those of their pages and bindings. Under homotopy theoretic conditions on the monodromy we prove an integral decomposition result for the based loop space on an open book, and under more relaxed conditions prove a rational loop space decomposition. The latter case allows for a rational dichotomy theorem for open books, as an extension of the classical dichotomy in rational homotopy theory. As a direct application, we show that for Milnor's open book decomposition of an odd sphere with monodromy of finite order the induced action of the monodromy on the homology groups of its page cannot be nilpotent.


Introduction
Open book decomposition is a convenient way to study manifolds.The purpose of this paper is to gain insight into the homotopy theory of open books by studying their based loop spaces.This fits into a larger program aimed at establishing homotopy theoretic properties of manifolds, and more generally, Poincaré Duality complexes [BB,BW,BT1,BT2,H,HT1,HT2,HT3,T].
Let V be a smooth compact (n − 1)-manifold with ∂V = ∅.Let h be a self diffeomorphism of V which restricts to the identity on ∂V .Let V h be the mapping torus of h, defined as the quotient space V h = (V × I)/ ∼ where I = [0, 1] is the unit interval and (v, 0) ∼ (h(v), 1).This has boundary ∂V × S 1 and projection to the second coordinate induces a fibre bundle Following the notation in [BC], write (∂V × D 2 ) ∪ id V h for the union of ∂V × D 2 and V h over the common subspace ∂V × S 1 .A closed n-manifold M is an open book if there is a diffeomorphism (2) for some V and h as above.The map h is called the monodromy of the open book.
Open books are of important interest in both topology and geometry.For instance, the fundamental open book theorem of Winkelnkemper [W] states that a simply-connected manifold of dimension greater than 6 is an open book if and only if its signature is 0. Classical work of Milnor [Mil] provides explicit open book decompositions of odd dimensional spheres.Other historical applications of open books can be found in [Ran, Appendix].More recently, Gitler and Lopez de Medrano [GL] used open books as a tool to show that certain families of manifolds arising from combinatorial constructions are diffeomorphic to connected sums of products of spheres.In the context of contact geometry, a remarkable work by Giroux [Gi] showed that a contact manifold admits an open book structure that is compatible with the contact structure.Very recently, Bowden and Crowley [BC] gave a topological obstruction to the existence of an open book structure on a contact manifold that has flexible pages.
There are other equivalent descriptions of open books.Up to homeomorphism, the open book M can be obtained from V h by identifying (x, t) with (x, s) for each x ∈ ∂V and t, s ∈ S 1 .For t ∈ S 1 the fibre π −1 (t given a condition on the diffeomorphism h.Recall that the loop space ΩX of a based topological space X is the space of all pointed, continuous maps from the circle into X, while the suspension ΣX is the double cone of X. Theorem 1.1.Let M be a path-connected open book for which there is a diffeomorphism M ∼ = (∂V × D 2 ) ∪ id V h .Suppose that h ≃ id relative to ∂V .Then there is a homotopy equivalence where the space F is the homotopy fibre of the inclusion ∂V → V .Consequently, there is an isomorphism Theorem 1.1 is proved as a consequence of the following purely homotopy theoretic result, which is interesting in its own right and can be applied elsewhere.Recall that X ∧ Y is the smash product of two based spaces X and Y .
Theorem 1.2.Let A, B, C and D be path-connected spaces and suppose that there is a homotopy pushout Then there is a homotopy equivalence where F and G are the homotopy fibres of f and g respectively.
To deal with open books where the diffeomorphism h need not be homotopic to the identity map (3) Recall that a path-connnected space X is called nilpotent if its fundamental group π 1 (X) is a nilpotent group and it acts nilpotently on the higher homotopy groups π i (X) for i ≥ 2. Simplyconnected spaces, connected H-spaces and loop spaces are examples of nilpotent spaces.
and ∂V are path-connected and nilpotent, and the inclusion of the boundary Then there is a rational homotopy equivalence where the space F is the homotopy fibre of the inclusion ∂V → V .
The decomposition in Theorem 1.3 is a rational version of that in Theorem 1.1 under a looser condition on the monodromy h.In Theorem 1.3, the nilpotence conditions on V and ∂V are mild.
Any simply-connected space is nilpotent, and for a simply-connected open book of dimension greater than 6, Winkelnkemper [W] showed that both V and ∂V can be chosen to be simply-connected.
As an application of Theorem 1.3 we study the relationship between the rational homotopy groups of manifolds and their possible open book structures.In rational homotopy theory, there is a classical dichotomy characterizing rational spaces [FHT,page 452], [FOT,Section 2.5.3].
Any connected nilpotent space X with rational homology of finite type and finite rational Lusternik-Schnirelmann category is either: A connected finite dimensional CW -complex, for example, has finite Lusternik-Schnirelmann category.Since a smooth compact manifold can be given the structure of a finite dimensional CWcomplex, it has rational homology of finite type and finite rational Lusternik-Schnirelmann category.
Theorem 1.4.Let M be a path-connected n-manifold satisfying an open book decomposition M ∼ = (∂V ×D 2 )∪ id V h , where V and ∂V are path-connected and nilpotent, and the inclusion of the boundary ∂V −→ V induces an epimorphism of fundamental groups.Suppose that one of the following holds: (b) i * • e(h) m * = i * for some m ∈ Z + and the monodromy h acts nilpotently on the homotopy groups π * (V ).

Then either:
(1) M is rationally elliptic, in which case V is also rationally elliptic, the homotopy fibre of the inclusion ∂V ֒→ V is rationally homotopy equivalent to a sphere S l , and (2) M is rationally hyperbolic, in which case either V is rationally hyperbolic or the homotopy fibre of the inclusion ∂V ֒→ V is not rationally homotopy equivalent to a sphere.
In [BC] In contrast, our conditions in parts (a) and (b) of Theorem 1.4 for e(h) * on the rational homotopy groups can be viewed as a sort of homotopy order condition on the monodromy.The condition in part (a) is satisfied, for example, if the monodromy h is homotopic to the identity and the condition in part (b) is satisfied, for example, if h is of finite order.Viewed this way, Theorem 1.4 gives a rational dichotomy of open books with a homotopy order condition.Having one of the conditions in parts (a) and (b) hold is necessary, this is illustrated in Proposition 6.3 using Milnor's open book decompositions of odd spheres.Further, the nilpotent action condition for the monodromy in case (b) is necessary as illustrated in Example 6.5.In Section 6, we also compare Theorem 1.4 with a result of Grove and Halperin [GH] on the rational ellipticity of double mapping cylinders.
It is worth pointing out that, as methods are homotopy theoretical, the salient points in the arguments are not that V is a manifold and h is a diffeomorphism but that the inclusion ∂V −→ V is not a rational equivalence and h is a homotopy equivalence.
The paper is organized as follows.In Section 2 we prove the general result Theorem 1.2, give an integral loop space decomposition for open books with monodromy homotopic to the identity map, and prove Theorem 1.1.In Section 3 we give a loop space decomposition of the double of The authors would like to thank one referee for pointing out the necessity of the nilpotence condition in Theorem 1.4 and for Example 6.5, and are indebted to another referee for many valuable comments, including suggesting Theorem 1.2 as a means to prove Theorem 1.1.

An integral loop space decomposition of open books with homotopically trivial monodromy
As we will need to work with homotopy fibrations and homotopy groups, throughout it will be assumed that all spaces and maps are pointed.In particular, if V is a compact manifold with boundary ∂V = ∅ then assume that a basepoint v 0 for V has been chosen that is also in ∂V .
We begin by proving the general result, Theorem 1.2.Let I = [0, 1] be the unit interval with 0 as basepoint.For path-connected spaces X and Y , the (reduced) join is the quotient space where (x, 0, y) ∼ (x ′ , 0, y), (x, 1, y) ∼ (x, 1, y ′ ) and ( * , t, * ) ∼ ( * , 0, * ) for all x, x ′ ∈ X, y, y ′ ∈ Y and Proof of Theorem 1.2.First observe that there is a pushout map (4) for some map θ.Let H be the homotopy fibre of θ.Pulling back H −→ Q with each of the maps in the homotopy pushout defining Q then gives a homotopy commutative cube in which the bottom face is a homotopy pushout, the four sides are homotopy pullbacks, and the maps a and b are induced maps of fibres.Mather's second cube theorem (Theorem A.2) implies that the top face is a homotopy pushout.
We now identify the homotopy classes of a and b.The rear face of the cube is the left square in the homotopy fibration diagram where the * in the lower left corner has been added to make clear the lower row is a product of homotopy fibrations.Observe that the entire diagram is a product of two homotopy fibration diagrams, one for the left factors and one for the right factors.This implies that a is the product The homotopy pushout in the top face of (5) therefore implies that H is homotopy equivalent to the pushout of the projections π 1 and π 2 , which is homotopy equivalent to F * G. Therefore, we obtain of the wedge into the product.By [Ga], this inclusion has a right homotopy inverse after looping, implying that Ωθ has a right homotopy inverse.Thus the homotopy fibration splits after looping to give a homotopy equivalence ΩQ ≃ ΩC × ΩD × Ω(F * G).
To apply Theorem 1.2 in the context of open books a lemma is first required.Let ι : ∂V −→ V be the inclusion of the boundary.
Lemma 2.1.Let V be a smooth compact (n − 1)-manifold with nonempty boundary and h a selfdiffeomorphism of V that restricts to the identity on ∂V .If h ≃ id relative to ∂V then there is a Proof.Recall that V h = (V × I)/ ∼ where (v, 0) ∼ (h(v), 1) and there is a fibre bundle and π( [v, t]) = e 2πit is the projection to the second factor.Since id ≃ h −1 relative to ∂V , there is a homotopy giving the asserted commutative diagram.
We can now prove a homotopy decomposition for the based loops on a family of open books.
relative to ∂V , so by Lemma (2.1) the space V h may be replaced up to homotopy equivalence with V × S 1 and in a way that is compatible with the inclusion of the boundary ), that is, there is a homotopy pushout ( 6) where j is the standard inclusion.By Theorem 1.2, there is a homotopy equivalence where F and G are the homotopy fibres of ι and j respectively.As D 2 is contractible we obtain Proof.Collapsing the cylinder that there is a pushout, up to homotopy equivalences, where the maps  i for i = 1, 2 are the inclusions into the top and bottom copies of V .
Compose each of the four corners of the pushout with the folding map p and take homotopy fibres.Noting that p •  1 and p •  2 are both the identity map on V , we obtain homotopy fibrations that define the spaces P and F and the maps f and f .In each of these homotopy fibrations the map from the total space to the base factors through the map DV p −→ V , so we obtain a homotopy commutative cube where i is the inclusion DV = ∂(V × I) ֒→ V × I, and j 1 and j 2 are the induced injections.Since i is a cofibration, the pushout ( 8) is also a homotopy pushout.
In this section, we study the loop space homotopy type of M via (8) with a looser condition on the monodromy.To do so we have to pass to rational homotopy.Some lemmas are needed to prepare the way.Recall that P is the homotopy fibre of the folding map p : DV → V .Recall that F is the homotopy fibre of the inclusion of the boundary ι : an isomorphism on π k for k < m and an epimorphism on π m .Lemma 4.2.If V and ∂V are path-connected and nilpotent, and the inclusion ∂V −→ V is 1connected, then F is path-connected, P ≃ ΣF is simply-connected, and both DV and M are pathconnected and nilpotent.
Proof.Since ∂V −→ V is 1-connected, it induces an isomorphism on π 0 and an epimorphism on π 1 .
The long exact sequence of homotopy groups for the homotopy fibration F −→ ∂V ι −→ V then implies that F is path-connected.By Lemma 3.1, P ≃ ΣF , implying that P is simply-connected.
The homotopy fibration P → DV p → V then implies that DV is path-connected.As in the proof of Lemma 3.1, there is a homotopy pushout In general, Rao [Rao,Theorem 2.1] shows that if W is the homotopy pushout of maps where X, Y and Z are all nilpotent and f and g induce epimorphisms in π 1 , then W is nilpotent.In our case, both ∂V and V are path-connected and nilpotent and ι induces an epimorphism on π 1 , so DV is nilpotent.
By Lemma 4.1 there is a homotopy fibration P −→ DV i −→ V × I.As P ≃ ΣF is simplyconnected, the long exact sequence of homotopy groups for the fibration implies that In the pushout (8), V × I is path-connected and nilpotent by hypothesis, we have seen that DV is path-connected and nilpotent, and as i is 2-connected both i and i • e(h) induce epimorphisms in π 1 .Therefore, by [Rao,Theorem 2.1], the pushout M is path-connected and nilpotent.
We can now prove Theorem 1.3.
Proof of Theorem 1.3.The plan is to construct a rational homotopy fibration ΩV where τ is null homotopic, and use Lemma 4.2 to identify P as ΣF .To do this the first cube theorem (Theorem A.1) will be used to produce a map ΣP −→ M with the right homotopy fibre.
To begin, we compare the homotopy fibres of the maps DV where e ′ is an induced map of fibres.Since e(h) is a diffeomorphism it induces an isomorphism of homotopy groups.The map of long exact sequences of homotopy groups induced by the map of fibrations (9) and the five lemma therefore imply that e ′ * : π * (P ′ ) → π * (P ) is an isomorphism.In particular, as P is simply connected by Lemma 4.2, P ′ is simply-connected.Since V is nilpotent by hypothesis, P and P ′ are simply-connected, and DV is nilpotent by Lemma 4.2, we can consider the rationalization of Diagram (9).Note the isomorphism e ′ * also implies that for each Moreover, since V is compact it is of finite type, and therefore so are DV and V × I.This implies that both sides of (10) are finite.
By assumption, i * • e(h) * = i * on rational homotopy groups.Therefore the composite . By Lemma 4.2, P ≃ ΣF , so P is a wedge of simply-connected spheres rationally.Thus the triviality of ( 11) implies that the composite • g is rationally null homotopic.Hence, rationally, there is a lift e ′′ of g to the homotopy fibre of i • e(h), giving a homotopy commutative diagram ( 12)

Consider again the homotopy fibration
The map i has a right inverse up to homotopy.This is because the composite base of the cylinder, where  1 is the inclusion into the base of DV , and this composite of inclusions is a homotopy equivalence since I is contractible.Thus g * is injective on homotopy groups.The homotopy commutativity of (12) therefore implies that e ′′ is injective on rational homotopy groups.
The equality in (10) and the fact that both sides of that equality are finite then imply that e ′′ * is an isomorphism on rational homotopy groups.It follows by Whitehead's Theorem that e ′′ is a rational homotopy equivalence.Hence, rationally, there is a homotopy fibration P The nilpotence conditions in Lemma 4.2 imply that we may rationalize all spaces and maps to consider the cube where CP is the rationalization of the (reduced) cone on P and both Q and g will be defined momentarily.Since there are rational homotopy fibrations − − → V × I, the composites i • g and (i • e(h)) • g are null homotopic.The (pointed) homotopy P × I −→ V × I which at time 0 is i • g and time 1 is the constant map implies that there is a map CP −→ V × I which makes the left face of (13) strictly commute.Similarly, there is a map CP −→ V × I which makes the rear face of (13) strictly commute.Note the two maps from CP to V × I may be different if the homotopies are different.The bottom face of (13) will strictly commute once we replace M by the point-set pushout of the rational maps i and i • e(h), which is rationally homotopy equivalent to M .Hence for convenience we may assume the bottom face of ( 13) is a point-set pushout.Define Q as the point-set pushout of P −→ CP with itself, so the top face strictly commutes.The strict commutativity of the left face, rear face and bottom face, and the fact that Q is a point-set pushout, implies that there is a pushout map Q g −→ M that makes the front and right faces of (13) strictly commute.Therefore the cube (13) strictly commutes, the bottom and top faces are pushouts, and the left and rear faces are homotopy pullbacks since CP is contractible and there are rational homotopy fibrations − − → V × I. Hence, by Theorem A.1, the front and right faces in (13) are also homotopy pullbacks.
Finally, we draw consequences.Observe that as CP is contractible, the pushout in the top face of (13) implies that Q ≃ ΣP .Since the right face is a homotopy pullback, we obtain a diagram of rational homotopy fibrations implying that τ is null homotopic.Thus there is a rational homotopy equivalence ΩM ≃ Q ΩV ×ΩΣP , and Lemma 3.1 then refines this to a rational homotopy equivalence ΩM ≃ Q ΩV × ΩΣ 2 F .

The proof of Theorem 1.4
In this section, we prove Theorem 1.4.
By Theorem 1.3 there is a rational homotopy equivalence ΩM ≃ Q ΩV × ΩΣ 2 F , which implies that Recall that F is the homotopy fibre of the inclusion ι : ∂V → V .By assumption, V and ∂V are nilpotent and the inclusion of the boundary ∂V −→ V is 1-connected.Thus F is path-connected by Lemma 4.2, and then [HMR,Chapter II,Proposition 2.13] implies that F is also nilpotent.Since V is a smooth compact (n − 1)-manifold, we have H n−1 (V, ∂V ; Z) ∼ = Z.This implies that ι is not a rational homotopy equivalence.In particular, F is not rationally contractible.Therefore Σ 2 F is not rationally contractible, in which case it is rationally a wedge of spheres.This implies that Σ 2 F is rationally hyperbolic unless it is a single sphere, in which case F ≃ Q S l for some l.It follows that M is rationally elliptic if and only if V is rationally elliptic and F ≃ Q S l for some l.In this case Otherwise M is rationally hyperbolic, and this happens if and only if either V is rationally hyperbolic or F ≃ Q S l for any l.This proves the theorem in the case (a) when i * • e(h) * = i * .
Next, we prove case (b).By assumption, there is an m ∈ Z + such that i * • e(h) m * = i * .Notice that, by definition of e(h), we have e(h) m = e(h m ).Therefore i * • e(h m ) * = i * , so by the special case the theorem holds for the open book On the other hand, by definition of V h as a quotient space there is a quotient map c m is well-defined and continuous.Further, since we see that c m factors through q h m to define a map By its construction, c m covers the standard m-sheeted covering t m : S 1 → S 1 defined by t m (z) = z m for any z ∈ C, and by (1) there is a morphism of fibre bundles (14) Further, since h restricts to the identity map on ∂V , the m-sheeted covering c m restricts to id × t m on ∂V × S 1 .
In order to apply rational homotopy theory to Diagram ( 14), we need to prove that the manifolds V h and V h m are nilpotent spaces.Indeed, for the fibre bundle (1) of the mapping torus V h , the canonical action of π 1 (S 1 ) ∼ = Z on the homotopy groups π * (V ) of the fibre is determined by its restriction on a generator of π 1 (S 1 ), which is the monodromy action h * : π * (V ) −→ π * (V ).By assumption in case (b), this action is nilpotent.Moreover, the fibre bundle implies that π for any i ≥ 2, and up to homotopy equivalence V is the universal covering of V h .It follows that the action of the fundamental group π 1 (V h ) on the higher homotopy groups π i (V h ), i ≥ 2 can be identified with the action of π 1 (S 1 ) on the homotopy groups π i (V ), which is nilpotent by the previous discussion.Hence, the manifold V h is a nilpotent space.This implies that the m-sheeted covering V h m is also a nilpotent space by [Mis,Theorem 1.3].In particular, rationalizations of V h and V h m exist.Now we draw consequences.First, since the map t m induces an isomorphism on rational homotopy groups, the five-lemma applied to the morphism of fibre bundles ( 14) implies that c m also induces an isomorphism on rational homotopy groups.Whitehead's Theorem therefore implies that c m induces a rational homotopy equivalence.Second, the fact that c m restricts to id × t m on ∂V × S 1 implies that there is a map such that Φ restricts to c m on V h m and to (id × t m ) on (∂V × D 2 ).As all spaces involved are nilpotent, it follows that Φ is also a rational homotopy equivalence and M ≃ Q M ′ .Therefore, as the theorem holds for M ′ , it also holds for M .This completes the proof of the theorem.

The necessity of the homotopy conditions on the monodromy
In this section we investigate the homotopy conditions on the monodromy in Theorem 1.4 further.
First is a comparison with work of Grove and Halperin and second is an application to Milnor's open books.
In [GH], Grove and Halperin showed that if there are maps φ i : X → B i (i = 0, 1) whose homotopy fibre are rationally spheres then the double mapping cylinder DCyl(X) := B 0 ∪ φ0 (X × I) ∪ φ1 B 1 is rationally elliptic.In our case, the open book M ∼ = (∂V × D 2 ) ∪ id V h is a double mapping cylinder via the homeomorphism where j : S 1 ֒→ D 2 is the canonical inclusion and J : ∂V × S 1 ֒→ V h is the restriction of the mapping torus to the boundary ∂V .It is clear that the homotopy fibre of (id implies that the left square is a pullback, and therefore the homotopy fibre of J is the same as that of the inclusion ∂V ֒→ V .When the latter fibre is rationally a sphere, Grove and Halperin [GH,Corollary 6.1] showed that M is rationally elliptic if and only if ∂V is.In particular, this is consistent with part (1) of Theorem 1.4.The classification result in Theorem 1.4 can therefore be thought of extending Grove and Halperin's work in the elliptic case.
Example 6.1.Let N × S m+1 be the product of a closed manifold N and the (m + 1)-sphere with with page N ×D m and trivial monodromy.Note that the homotopy fibre of the inclusion N ×S m−1 ֒→ N × D m is S m−1 .Then the conclusion of Theorem 1.1 reduces to the obvious homotopy equivalence Further, the conclusion of Theorem 1.However, for m ≥ 4 there could be such constructions.Indeed, for each m ≥ 5 Cerf [Ce] proved that where Θ m+1 denotes the group of oriented homotopy (m+ 1)-spheres up to oriented diffeomorphism.
As Kervaire and Milnor [KM] showed that Θ m+1 is always finite, the monodromy h of the open book construction is of finite order up to isotopy.Since the page D m is contractible, the induced homomorphism is trivial, and the homotopy fibre of the inclusion S m−1 ֒→ D m is S m−1 , Theorem 1.4 applies to show that S h is rationally elliptic and In particular, S h is rationally a sphere.
For m = 4, the famous recent work of Watanabe [Wa] disproved the conjecture that Diff ∂ D 4 is contractible.However, as in [Wa,Remark 1.2], information about π 0 (Diff ∂ D 4 ) is still unknown.
Nevertheless, as above, any class in π 0 (Diff ∂ D 4 ) will still give an open book structure on a corresponding rational sphere.
Theorem 1.4 also has an interesting application in the hyperbolic case.Consider Milnor's open book decompositions of spheres [Mil].Let f : C n+1 → C be a non-constant complex polynomial in z = (z 1 , . . ., z n+1 ), and Z = f −1 (0) be the algebraic set of zeros.Suppose z 0 ∈ Z is an isolated critical point of f and there is a diffeomorphism S ǫ ∼ = S 2n+1 , where S ǫ is a sphere of radius ǫ centered at z 0 for some sufficiently small ǫ.Then there is a compact 2n-manifold where the positive integer µ is the multiplicity of z 0 as solution to the system of polynomial equations {∂f /∂z j = 0} n+1 j=1 .
Proposition 6.3.Let h be the monodromy of the open book S 2n+1 determined by an isolated critical point z 0 of a complex polynomial f : C n+1 → C with n ≥ 3.If the multiplicity of z 0 as solution to the system of polynomial equations {∂f /∂z j = 0} n+1 j=1 is greater than 1, then either: • there is no integer m such that h m is rationally homotopic to the identity map, or • the monodromy h acts non-nilpotently on the homology groups H * (V ; Z).
Proof.To obtain a contradiction assume that h m is homotopic to the identity map and the monodromy h acts nilpotently on the homology groups H * (V ; Z).By a similar argument to that in the proof of Theorem 1.4, the latter condition means that π 1 (V h ) acts nilpotently on the homology groups H * (V ) of the universal cover.It follows that V h is a nilpotent space by [HMR, Chapter II, Remark 2.19], which implies that h acts nilpotently on the homotopy groups π * (V ).
Further, as n ≥ 3, the binding ∂V is simply-connected by [Mil,Proposition 5.2].For the page, by (15) V ≃ µ i=1 S n and it is rationally hyperbolic as µ ≥ 2 by hypothesis.Therefore Theorem 1.4 can be applied and it implies that the associated open book S 2n+1 = (∂V × D 2 ) ∪ id V h is rationally hyperbolic.This contradicts the fact that spheres are rationally elliptic, and hence the proposition follows.
Notice too that for the open book in Proposition 6.3 with µ ≥ 2, the homotopy fibre of ∂V ֒→ V cannot be a sphere rationally.Otherwise, by the result of Grove and Halperin [GH,Corollary 6.1] ∂V is rationally elliptic as S 2n+1 is, and then so is V .However, this contradicts (15) when µ ≥ 2.
We end this section with two examples.
The original statement of the second cube theorem [M, Theorem 25] assumes that there is a cube that homotopy commutes in which the bottom face is a homotopy pushout, the four sides are homotopy pullbacks, and there are compatibilities among the homotopies, and concludes that the top face is a homotopy pushout.The hypothesis that the homotopy commutativity of the cube is due to it being induced by taking fibres over a map D h −→ Z lets one bypass the compatibilities.
a codimension 1 submanifold of M whose image is the t-th page of the open book.And the image of ∂V × S 1 in M is a closed codimension 2 submanifold called the binding of the open book.This description is the reason behind the name "open book".Our first result establishes an integral homotopy decomposition for the based loops on open books relative to ∂V we turn to rational homotopy theory.The argument will involve another equivalent description of open books; see for instance [Q, BC].Let DV = ∂(V × I) be the (trivial) double of V and let i : DV −→ V × I be the inclusion of the boundary.Define a self-diffeomorphism e(h) : DV −→ DV that extends h by e(h)(v, 0) = v, e(h)(v, t) = (v, t) if v ∈ ∂V and e(h)(v, 1) = h(v).Let (V × I) ∪ e(h) (V × I) be the manifold obtained by gluing together the image of i in the left copy of V × I and the image of e(h) • i in the right copy of V × I.The open book M in (2) can be expressed as a twisted double via a diffeomorphism , Bowden and Crowley proved that if M is a contact manifold admitting an open book structure whose pages are flexible Weinstein manifolds, then the map e(h) : DV e(h) −→ DV has the property that e(h) * : H * (DV ; Z) → H * (DV ; Z) is the identity map up to half the dimension of V .

a
manifold with boundary.We then turn to the rational homotopy of open books.In Section 4 we consider a special case of the homotopy order condition for open books, give a rational loop space decomposition for such open books, and prove Theorem 1.3.Section 5 is devoted to proving the dichotomy result, Theorem 1.4, and Section 6 gives an example proving the necessity of the homotopy order condition in Theorem 1.4.Since our techniques in homotopy theory are based on the two cube theorems of Mather [M], we state and comment on them in Appendix A for those interested readers from manifold topology and geometry.Acknowledgements.The first author was supported in part by the National Natural Science Foundation of China (Grant nos.12331003 and 12288201), the National Key R&D Program of China (No. 2021YFA1002300), the Youth Innovation Promotion Association of Chinese Academy Sciences, and the "Chen Jingrun" Future Star Program of AMSS.The authors would like to thank Xiaoyang Chen for suggesting a study of the rational ellipticity of open books and for related discussions, and Zhengyi Zhou for helpful discussions on Milnor fibrations.

3.
A loop space decomposition of the double of V By definition, the double of V is DV = ∂(V × I).Define the folding map p : DV → V by p(v, 0) = v, p(v, t) = v for v ∈ ∂V and p(v, 1) = v.Let P be the homotopy fibre of p. Lemma 3.1.There are homotopy equivalences ΩDV ≃ ΩV × ΩΣF and P ≃ ΣF.
which the bottom face is a homotopy pushout and the four sides are homotopy pullbacks.Mather's second cube theorem (Theorem A.2) implies that the top face is a homotopy pushout.In particular, the top face of the cube being a homotopy pushout implies that P ≃ ΣF while the right face of the cube being a homotopy pullback implies that there is a homotopy fibration diagram ΩV The right square implies that the connecting map δ is null homotopic.It follows that there is a homotopy equivalence ΩDV ≃ ΩV × ΩP ≃ ΩV × ΩΣF, proving the lemma.4. A rational loop space decomposition of certain open books By (3) there is a pushout diagram (8)

Lemma 4 . 1 .
The homotopy fibre of the inclusion i : DV → V × I is homotopy equivalent to P .Proof.The projection V ×I π −→ V is a homotopy equivalence, so the homotopy fibre of i is homotopy equivalent to the homotopy fibre of π • i.But π • i = p and, by definition, the homotopy fibre of p is P .
× I where g is a name for the map from the fibre to the total space.Let P ′ is the homotopy fibre of i • e(h) : DV → V × I. Then there is a pointwise.It is known that Diff ∂ D m is contractible for m = 1, 2 and 3: the case m = 1 is easy, the case m = 2 was proved bySmale [S], and the case m = 3, known as the Smale conjecture, was proved by Hatcher[Hat].In particular, when m = 2 or 3 we cannot construct an open book with page D m and nontrivial monodromy.
with boundary and an abstract open book decomposition of S 2n+1 via diffeomorphismsS 2n+1 ∼ = S ǫ ∼ = (∂V × D 2 ) ∪ id V h ,where the binding is ∂V ∼ = Z ∩ S ǫ and the interior (V − ∂V ) of the page V is diffeomorphic to the fibre of the Milnor fibrationφ : S ǫ − Z −→ S 1 defined by φ(z) = f (z) ||f (z)|| .Milnor showed that there is a homotopy equivalence 4 reduces to stating that N × S m+1 is rationally elliptic if and only if N × D m is, which automatically holds as S m+1 is rationally elliptic.Example 6.2.Following Example 6.1, we can try to construct examples with nontrivial monodromies, for which we need to choose nontrivial homotopy classes of elements in the diffeomorphism group Diff ∂ D m , the group of self-diffeomorphisms of D m which fix a neighborhood of ∂D m