aa r X i v : . [ m a t h . M G ] D ec Efficient Cycles in Loop Space
Robin Elliott
Abstract
This paper investigates how the geometry of a cycle in the loop spaceof Riemannian manifold controls its topology. For fixed β ∈ H n (Ω X ; R )one can ask how large |h β, Z i| can be for cycles Z supported in loops oflength ≤ L and of volume ≤ L n − for a suitably defined notion of volumeof in loop space. We show that an upper bound to this question providesupper bounds Gromov’s distortion of higher homotopy groups. We alsoshow that we can exhibit better lower bounds than are currently knownfor the corresponding questions for Gromov’s distortion. Specifically, weshow there exists a β detecting the homotopy class of the puncture in[( CP ) × S ] ◦ and a family of cycles Z L with the geometric boundsabove such that |h β, Z i| = Ω( L / log L ). The purpose of this paper is to investigate the existence of cycles in loop spacewith small volume, small suplength and large homological degree. If Z is a chainin the loop space Ω X of a metric space X , we say that suplength( Z ) ≤ L if Z is supported in the space Ω ≤ L X of loops in X of length at most L . The notionof volume of a chain will be defined using the n -dimensional Hausdorff measureof the chain, with respect to a metric on Ω X induced by the metric on X . Thehomological degree of a cycle will be measured by evaluating it against a fixedcohomology class β ∈ H n (Ω X ). We work with real coefficients throughout thepaper unless otherwise stated.More precisely, given β ∈ H n (Ω X ) define the cohomological distortion of βδ β ( L ) = sup {h β, Z i | Suplength( Z ) ≤ L and Vol( Z ) ≤ L n } and investigate the asymptotic growth of δ β ( L ) as L → ∞ . Here the volumebound of ≤ L n is chosen so that this notion of distortion matches existingnotions of distortion, as described in Section 1.1. Example 1.
Consider β a generator of H n − (Ω S n ) ∼ = R for n ≥
2. This β detects the degree of a map f : S n → S n , in the sense that if ˆ f : S n − → Ω S n is adesuspension of f (under the suspension-loop adjunction) then h β, ˆ f ∗ [ S n − ] i = C deg( f ) for some nonzero constant C depending on the choice of generator β .We will show that δ β ( L ) = Ω( L n ). 1et ζ n be a fixed cycle representing the generator of H n − (Ω S n ), i.e. ζ n isa sweepout of S n by loops. For L ∈ N , let { L } : Ω X → Ω X be the map thatsends a loop γ to its L -fold concatenation, γ · γ · . . . · γ . Once the relevant metricsetup has been established (see section 2), it can be shown that L n − { L } ∗ ζ n are a family of ( n − S n witnessing that δ β ( L ) & L n .Here is another construction of a different family giving the same bound.Let f L : S n → S n be a family of L -Lipschitz maps that are degree Θ( L n ).Then take the family of cycles to be (Ω f L ) ∗ ζ n . This gives the same asymptoticlower bound of L n for δ β ( L ). In [Ell20] it is shown that this bound of L n isasymptotically sharp. Example 2.
Consider β a generator of H n − (Ω S n ) ∼ = R for n ≥ β detects the Hopf invariant of a map S n − → S n in the same sense as above.We will show that its cohomological distortion is Ω( L n − ).Consider the same ζ n as in the previous example. The (2 n − ζ n , ζ n ]generates H n − (Ω X ), where here the bracket is the graded commutator as-sociated to the Pontryagin product on C ∗ (Ω X ). Then the family of cycles Z L := L n − [ { L } ∗ ζ n , { L } ∗ ζ n ] witnesses that δ β ( L ) & L n .Alternatively, we could instead take a family of maps f L : S n − → S n that are L -Lipschitz and have Hopf invariant Θ( L n ). The same construction Z L := (Ω f L ) ∗ ζ n − as in the previous example also gives the cohomologicaldistortion of β is & L n . In [Ell20] it is shown that this bound of L n isasymptotically sharp. The notion of cohomological distortion is related to Gromov’s notion of homo-topical distortion, which has its origins in [Gro78]. We reformulate Gromov’snotion, following [BM19]. Given a rational homotopy class α ∈ π n ( X ) ⊗ Q , the distortion δ α ( L ) of α is δ α ( L ) = sup { k | there exists an L -Lipschitz map S n → X with [ f ] = kα } The relationship between the two notions of distortion comes from the fol-lowing observation. There is a constant
C > L -Lipschitz map f : S n → X induces an ( n − b f ∗ [ S n − ] in Ω X that is suplength at most CL and volume at most CL n . This gives the following proposition. Proposition 1.1.
Let β ∈ H n − (Ω X, R ) and α ∈ π n ( X ) ⊗ Q . The dualof the Hurewicz homomorphism defines a real homotopy functional τ ∨ ( β ) ∈ Hom ( π n ( X ); R ) . If h τ ∨ ( β ) , α i 6 = 0 then δ β ( L ) & δ α ( L ) . Note that bound of L n on the n -volume of Z in the definition of cohomolog-ical distortion was chosen so the inequality in Proposition 1.1 does not have tocontain any additional factors of L .In [Gro99] Gromov conjectured that the distortion of a rational homotopyclass α ∈ π n ( X ) ⊗ Q was determined by the minimal model of X . In the caseof the least distorted homotopy elements, a strong interpretation of this can bestated as follows. 2 onjecture 1.1. (Gromov) The homotopical distortion of a rational homotopyclass α ∈ π n ( X ) ⊗ Q is Θ( L n ) if and only if the Hurewicz image of α is nonzero.Otherwise, the homotopical distortion is Ω( L n +1 ) . Recently in [BM19] Manin and Berdnikov disproved the above conjecture.Specifically, they exhibited a punctured 6-manifold Y = [( CP ) × S ] ◦ andshow the class of the puncture α ∈ π ( Y ) ⊗ R is in the kernel of the Hurewiczhomomorphism but has homotopical distortion o ( L ). The best current lowerbound on the distortion of α is Ω( L ), given by composing α with degree L L -Lipschitz self maps of S n . In this paper we investigate the homotopy class from the end of previous sectionunder the lens of cohomological distortion.
Theorem A.
Let Y = [( CP ) × S ] ◦ and α ∈ π ( X ) ⊗ Q be the class of thepuncture. Then there exists a β ∈ H (Ω Y ) such that its image under the dualHurewicz detects α and such that β has cohomological distortion Ω( L / log L ) . Theorem A is proved by explicitly constructing a β in terms of iteratedintegrals, and explicitly constructing an efficient family of 4-cycles Z L on Ω Y . Atthe heart of the construction of Z L is the construction of the following efficient(small volume) homology which exists in the loop space of any Riemannianmanifold X , not just the Y stated above. Theorem B.
Let X be a Riemannian manifold and let L > be a power oftwo. For any n ≥ , fix n -cycles [ S n ] representing the fundamental class of S n .Let f : S n +1 → X and f : S n +1 → X be Lipschitz maps. For i = 1 , let Z i be the n i -cycle ( b f i ) ∗ [ S n i ] in Ω X , where b f i is the desuspension of f i . Let [ · , · ] denote the graded commutator associated to the Pontryagin product on chainson Ω X . Then the two homologous ( n + n ) -cycles [ { L } ∗ Z , { L } ∗ Z ] and L { L } ∗ [ Z , Z ] admit a homology P = P ( f , f , L ) with suplength at most CL and volume CL log L . Here C > is a constant depending only on n i , the Lipschitz constantsof the maps f i , and the choice of metric on S n i . In particular, C is independentof L and X . Moreover, if the image of f i lie in some skeleton X ( k ) of X then P can be taken to be supported in the subspace Ω X ( k ) ⊂ Ω X . The punchline of Theorem B is that we can find a . L -suplength homologybetween these two cycles with volume at most a constant times L log L . Aweaker bound of L is easier to derive, but then only gives a lower bound ofthe cohomological distortion of β ∈ H (Ω Y ) of L . This is the same boundthat comes from the trivial bound on the homotopical distortion of α . It is notcurrently known if Theorem B could be improved to give a linear bound on thevolume of Y . If so, it would give a lower bound on the cohomological distortionof β of L , strictly asymptotically larger than Berdnikov and Manin’s upperbound of o ( L ) for the homotopical distortion of α .3 .3 Outline of the paper In Section 2, we set up the geometric and metric notions needed in the paper.Section 3 contains the proof of Theorem B, i.e. the construction of a small-volume homology between a certain pair of homologous cycles in Ω X for any X . Section 4 contains the proof of Theorem A: the construction of the familyof cycles in Ω Y for Y = [( CP ) × S ] ◦ and the the geometric and topologicalbounds on these cycles. I would like to thank my advisor, Larry Guth, for constant advice and enthu-siasm. I would also like to thank Fedya Manin for helpful conversations andexplanations; much of this work is inspired by the content and questions in[Man19] and [BM19]. I also benefited from Dev Sinha’s seminar at MIT inSpring 2020 and the geometric insights exposited there. Finally I would like tothank Haynes Miller and Luis Kumandari for helpful conversations and usefulcomments.
Let X be a Riemannian manifold with basepoint x . By Ω X we mean thespace of piecewise smooth Moore loops , i.e. a point in Ω X is a pair ( a, γ )with curfew a ≥ γ : [0 , a ] → X with γ (0) = γ ( a ) = x ∈ X . In this model Ω X is a strictly associative H -space with n -ary multiplication µ n : (Ω X ) n → Ω X given by summing of curfews and concatenation of loops.Let C ∗ (Ω X ) denote the (real) cubical chain complex. The concatenationmultiplication µ : Ω X × Ω X → Ω X induces a multiplication · : C m (Ω X ) ⊗ C n (Ω X ) → C m + n (Ω X )on chains. The graded commutator associated to this multiplication is given by[ Z , Z ] := Z · Z − ( − | Z || Z | Z · Z . Remark.
The notion of curfew is a technical requirement to ensure that theJacobi relation is satisfied on the nose for the Lie bracket that will be definedon C ∗ (Ω X ) below. We will always be considering cycles of constant curfew,although the size of this curfew may vary. The curfew is unimportant to thegeometry, and is also distinct from the much more important quantity suplength defined in Section 1.The distance metric on Ω X we will use is given by d Ω X (( a , γ ) , ( a , γ )) := | a − a | + sup t ∈ [0 , d X ( γ ( ta ) , γ ( ta )) . X gives an n -dimensional Hausdorff measure on Ω X foreach n . For a simplex σ in Ω X , define the n -volume of σ as the n -dimensionalHausdorff measure of its image. Given an arbitrary real chain c = P λ i σ i , definethe volume of c by P | λ i | Vol( σ i ).There are other ways to define the volume of a chain. For example, theRiemannian metric g on X gives Ω X the structure of an infinite-dimensionalFinsler manifold in the following way. Given a piecewise smooth loop γ ∈ Ω X with curfew a , the tangent space at γ , denoted T γ Ω X , is the space of piecewisesmooth vector fields along γ which vanish at the endpoints of γ . Given such avector field V , define the norm on V to be || V || ∞ = sup t ∈ [0 ,a ] || V ( t ) || ( X,g ) where the norm on the right hand side is from the Riemannian metric. Fromhere, if we assume our chain Z is a pseudomanifold then we can pull back theFinsler metric to Z and evaluate the volume of Z using this pullback metric.These two notions of volume agree, up to a constant. The important pointthat will be used in this paper is that given two chains Z and Z of dimensions m and n respectively, there exists a constant C depending only on the dimensionsof Z and Z such that Vol m + n ( Z · Z ) ≤ C Vol m ( Z )Vol n ( Z ).Of interest to us will be the map { L } : Ω X → Ω X given by (for any L ∈ N )the composite of the L -fold diagonal and L -fold multiplicationΩ X ∆ L −−→ (Ω X ) L µ L −−→ Ω X. On the constant-curfew subspaces of Ω X , { L } is 1-Lipschitz and so { L } ∗ is non-increasing on the volume of constant-curfew chains. It does howevermultiply the suplength of chains by a factor of L , and multiply the curfew ofpoints by a factor of L .Given two chains Z , Z with suplengths L , L and volumes V , V respec-tively, the chain [ Z , Z ] has suplength at most L + L and volume at most CV V for some constant C depending only on the dimensions of the chains.A spherical cycle is one that is equal to f ∗ [ S n ] for some map f : S n → Ω X and some cycle [ S n ] representing the fundamental class of S n . Such cycles are primitive (they evaluate to zero on all nontrivial cup products), but the converseis not true: for example the graded commutator [ Z , Z ] of two spherical is acycle that is primitive but not spherical. The two notions do agree on thelevel of homology: a primitive cycles is homologous to a spherical cycle by theMilnor-Moore theorem [MM65].Finally, recall Samelson’s theorem [Sam53]. Let τ : π n +1 ( X ) → H n (Ω X )be the Hurewicz map. Given α ∈ π n +1 ( X ) and α ∈ π n +1 ( X ) for n i ≥ α , α ] Wh ∈ π n + n +1 ( X ). Then τ [ α , α ] Wh = ( − n [ τ α , τ α ]where the bracket on the right hand side is the graded commutator in on H ∗ (Ω X ). 5 The proof of Theorem B
Here we restate Theorem B.
Theorem 3.1.
Let X be a Riemannian manifold and let L > be a power oftwo. For any n ≥ , fix n -cycles [ S n ] representing the fundamental class of S n .Let f : S n +1 → X and f : S n +1 → X be Lipschitz maps. For i = 1 , let Z i be the n i -cycle ( b f i ) ∗ [ S n i ] in Ω X , where b f i is the desuspension of f i . Then thetwo homologous ( n + n ) -cycles [ { L } ∗ Z , { L } ∗ Z ] and L { L } ∗ [ Z , Z ] admit a homology P = P ( f , f , L ) with suplength at most CL and volume CL log L . Here C > is a constant depending only on n i , the Lipschitz constantsof the maps f i , and the choice of metric on S n i . In particular, C is independentof L and X . Moreover, if the image of f i lie in some skeleton X ( k ) of X then P can be taken to be supported in the subspace Ω X ( k ) ⊂ Ω X .Proof. The L -Lipschitz condition on f and f implies that Z and Z havevolumes Vol( Z ) . L n and Vol( Z ) . L n respectively. To prove the theorem,we will construct pieces of P on many ”scales” and the final P will be the sumof the pieces. At a single scale we will construct the homology given in thefollowing lemma. Lemma 3.2.
There is a constant C = C ( n , n ) > such that the followingholds. Let Z , Z be constant-curfew spherical n - and n -cycles respectively withthe same curfew, which are the images of L Lipschitz maps b f i : S n i → Ω X , andsuch that Z and Z have suplength at most L . Then there is a constant-curfew ( n + n + 1) -chain P ′ = P ′ ( Z , Z ) with ∂P ′ = 2 { } ∗ [ Z , Z ] − [ { } ∗ Z , { } ∗ Z ] and P has suplength at most L and volume at most CL n + n ( L + L ) . Given the lemma, the homology P satisfying the theorem is built out of thepieces P ′ at each scale by: P := k − X i =0 i { i } ∗ P ′ ( { k − − i } ∗ Z , { k − − i } ∗ Z ) . In the i th summand, the chain P ′ ( { k − − i } ∗ Z , { k − − i } ∗ Z ) has volume atmost C k − i − L n + n +10 and suplength at most 4 · k − − i L . So each of the log L summands has suplength at most 4 LL and volume at most CLL n + n +10 , asrequired.It remains to prove the lemma. Proof. (of Lemma) Let the common constant curfew of Z and Z be a .6 otation. From now on, we will drop the · in the multiplication on chainsin Ω X and denote the multiplication by juxtaposition instead. We will denotethe 0-chain of a constant loop with curfew a by • . For example, Z and Z • aresimilar chains, but to obtain Z • from Z every point in the support of Z is ispost-concatenated with the constant loop of curfew a . In particular, Z and Z • have the same volume and suplength.We will construct the homology in the lemma in three parts:1. First, a homology P from [ { } ∗ Z , { } ∗ Z ] to an intermediate cycle Q .2. Next, a homology P from Q to another intermediate cycle Q . Thishomology will do nothing more than reparametrize loops in the supportof Q . However, it is this homology that contributes most of the volumeto P .3. Finally, a homology P from Q to 2 { } ∗ [ Z , Z ].To construct P , the crucial ingredient is the following. Recall that Z =( b f ) ∗ [ S n ], for b f : S n → Ω X . Moreover { } ∗ Z = µ ∗ ( b f × b f ) ∗ (Diag( S n ))where Diag( S n ) is the diagonal n -cycle ∆ ∗ [ S n ] in S n × S n and µ := µ :Ω X × Ω X → Ω X is loop concatenation. The cycle Diag( S n ) is homologousto the cycle Bouquet( S n ) = S n × {∗} ∪ {∗} × S n in S n × S n . Let Y S n be a fixed homology between these. Then we can push this forward into Ω X :we get the chain µ ∗ ( b f × b f ) ∗ ( Y S n ) which is a homology between { } ∗ Z and Z • + • Z .Similarly, we can fix an analogous ( n + 1)-chain Y S n and get a ( n + 1)-chain µ ∗ ( b f × b f ) ∗ ( Y S n ) is a homology between { } ∗ Z and Z • + • Z . Thedesired homology P is the following ( n + n + 1)-chain: P := [ { } ∗ Z , µ ∗ ( b f × b f ) ∗ ( Y S n )] + [ µ ∗ ( b f × b f ) ∗ ( Y S n ) , Z • + • Z ] . This is a homology between [ { } ∗ Z , { } ∗ Z ] and Q , where Q := [ Z • + • Z , Z • + • Z ] . The suplength of P is at most 4 L , as the suplength of each term in thebracket is at most 2 L . We also need to compute the volume of P . The maps b f and b f are L -Lipschitz, so similarly µ ◦ ( b f × b f ) and µ ◦ ( b f × b f ) are L -Lipschitz too. The chains Y S n and Y S n have fixed volumes C ( n ) and C ( n )independent of b f , b f and X . The cycle { } ∗ Z has volume equal to Vol( Z )and the cycle Z • + • Z has volume equal to 2Vol( Z ). Using [ Z ′ , Z ′′ ] hasvolume at most C Vol( Z ′ )Vol( Z ′′ ), the volume of P is at most CL n + n +10 .Next we construct P , and will save P to last. The idea is similar, exceptwe need to use the analogue of Y S n but for S n × S n and S n × S n in placeof S n and S n . In the first part we found a homology from Diag( S n ) toa linear combination of cycles representing the standard basis of H n ( S n × S n ) ∼ = H n ( S n ) ⊕ H n ( S n ). Here, we start with the diagonal ( n + n )-cycle7iag( S n × S n ) in ( S n × S n ) × ( S n × S n ). The group H n + n (( S n × S n ) × ( S n × S n )) has a basis represented by S n × S n × {∗} × {∗} , {∗} × S n × S n × {∗} ,S n × {∗} × {∗} × S n , {∗} × {∗} × S n × S n . Denote by Bouquet( S n × S n ) the sum of these that is homologous to Diag( S n × S n ), and let Y S n × S n be a homology between them.Now, { } ∗ [ Z , Z ] = ( µ ) ∗ ( b f × b f × b f × b f ) ∗ Diag( S n × S n ) − ( − n n ( µ ) ∗ ( b f × b f × b f × b f ) ∗ Diag( S n × S n )and so the ( n + n + 1)-chain in Ω XP := ( µ ) ∗ ( b f × b f × b f × b f ) ∗ Y S n × S n − ( − n n ( µ ) ∗ ( b f × b f × b f × b f ) ∗ Y S n × S n gives the required chain for the third part. This a homology between { } ∗ [ Z , Z ]and Q , where Q := ( Z Z • • + Z • • Z + • Z Z • + • • Z Z ) − ( − mn ( Z Z • • + Z • • Z + • Z Z • + • • Z Z )is the pushforward under the right products b f and b f of Bouquet( S n × S n )and Bouquet( S n × S n ).By the same argument as for P , the suplength of P is at most 2 L and thevolume of P is at most CL m + n +10 .Finally, we construct the homology P . Note that both Q (after expandingthe definition of the brackt) and Q are homologous to 4( Z Z ••− ( − n n Z Z •• ). Here the extra • s have been inserted (arbitrarily) to make this chain havecurfew 4 a , the same as Q and Q . We will construct the homology for thesecond part one summand at a time: for each summand of Q and Q we willexhibit a homology to either Z Z • • or Z Z • • .For example, let’s construct a homology between • Z Z • and Z Z • • . Let • a (Ω X ) a denote the subspace of Ω X consisting of loops of curfew 4 a whichare the constant loop when restricted to [0 , a ] ⊂ [0 , a ]. Consider the linearinterpolation homotopy H s : • a (Ω X ) a × I → Ω X that linearly interpolatesbetween H the identity map, and the map H that reparametrizes the loop γ in the following way. H ( γ : [0 , a ] → X ) = ( t γ ( t + a ) if t < at x if t ≥ a That is, if Z is a cycle with curfew 3 a then H send • Z to Z • . By ”lin-early interpolates” we mean that for s ∈ [0 , H s does the reparametriza-tion that linearly interpolates between the identity reparametrization and the8eparametrization in H . Note that H s preserves curfew for each s ∈ [0 , H induces a chain homotopy H ∗ such that H ∗ ( • Z Z • ) is a homologybetween • Z Z • and Z Z • • . Note that H ∗ does not increase suplength andfor any chain Z , Vol n +1 ( H ∗ ( Z )) . Suplength( Z )Vol n ( Z ) . All other summands are similar.The lemma follows, and so the theorem does too.
In this section we will build the efficient family of 4-cycles Z L on Ω Y for Y =[( CP ) × S ] ◦ . For L = 1, the cycle Z will be homologous (perhaps up torescaling) to τ ( α ), the Hurewicz image of a sweepout by loops of the homotopyclass of the puncture in Y . This will be done by showing that both [ Z ] and τ ( α ) lie in the 1-dimensional subspace of primitive elements in H (Ω Y ). For τ ( α ) this is immediate from the Milnor-Moore theorem; for Z it will followfrom the definition of Z outlined below.The cycles Z L will be built out of some building-block chains on Ω Y , thehomologies constructed above, and the bracket [ · , · ] on C ∗ (Ω Y ). The building-block chains on Ω Y , which we denote by A i , B , C i and D (1 ≤ i ≤
4) arethe generators of the Adams-Hilton construction applied to the natural celldecomposition of Y . We explicitly construct them below; see [AH55] for theconstruction in full generality.The natural cell structure on Y has five 2-cells: one for each of the 2-cellsin ( CP ) and one from the S . Let’s call these 2-cells ˜ A i for 1 ≤ i ≤ B respectively. Each of these 2-cells has trivial attaching map, so they can bethought of as embedded 2-spheres in Y . Take a constant-curfew sweepout of the2-sphere by loops, i.e. 1-cycle ζ in Ω S generating H (Ω S ) integrally. Thenpushing forward ζ under the loop of each of the five embeddings S ֒ → Y givesfive 1-cycles in Ω Y that we will denote by A i for 1 ≤ i ≤ B .The cell structure on Y also has five 4-cells: one for the product of the 2-cell in S and each 2-cell in ( CP ) , and one corresponding to the top cell of( CP ) (crossed with the 0-cell in S ). Let’s call these 4-cells ˜ C i for 1 ≤ i ≤ D respectively. We want to write down corresponding 3-chains in Ω Y , butthis is slightly trickier to do as these 4-cells have nontrivial attaching maps. Theattaching map of ˜ C i is the Whitehead product [ ˜ A i , ˜ B ] Wh . The attaching map φ D of ˜ D is homotopic to X i =1 [ ˜ A i , ˜ A i ] Wh Here, by slight abuse of notation, we conflate the embedded 2-sphere ˜ A with a map S → Y and similarly for the ˜ B i . D in Ω Y corresponding to ˜ D . Take a constant-curfew 3-chain ξ in Ω D which corresponds to a sweepout of loops of the 4-discrelative to its boundary; that is ∂ξ = ζ is a sweepout of Ω S . Pushing thisforward under the loop map of the inclusion of the 4-cell ˜ D gives us a 3-chain D ′ in Ω Y . The boundary of D ′ in Ω Y is a 2-cycle that is the sweepout of theattaching map φ D of ˜ D . Now by Samelson’s theorem [Sam53], ∂D is homologousto P i =1 [ A i , A i ], where now here the bracket denotes the commutator productin C ∗ (Ω Y ) rather than the Whitehead bracket. Fix a homology D ′′ betweenthem. Then our 4-chain D is the sum of D ′ and D ′′ .We proceed similarly to construct each C i . Sweeping out each 4-cell ˜ C i givesa 3-chain C ′ i with ∂C ′ i a 2-cycle in Ω Y that is a sweepout of [ A i , B ] Wh . Take ahomology C ′′ i in Ω Y from [ ˜ A i , ˜ B ] Wh to [ A i , B ]. Then define C i := C ′ i + C ′′ i .The details of doing this process in full generality for any cell complex X wereworked out by Adams and Hilton [AH55], who use the cell decomposition of X to give a small differential graded algebra that is quasi-isomorphic to C ∗ (Ω X ).However the full power of the Adams-Hilton construction is not needed heresince the cell decomposition of our Y was fairly simple: in particular Y ishomotopy equivalent to the cofiber of a map between wedges of spheres. Z L First we will define a cycle Z , which will then be modified to produce thecycles Z L . Given the 1-cycles A i and B and 3-chains C i and D constructed inthe previous section, Z is the cycle Z := [ B, D ] + 2 X i =1 [ A i , C i ] . The boundary of this is ∂Z = [ B, X i =1 [ A i , A i ]] + 2 X i =1 [ A i , [ B, A i ]]which vanishes due to the Jacobi identity applied to A i , A i and B .Now we define Z L . It will look similar to the definition of Z but witheach of the building-block chains A i , B , C i , D replaced by a modification ofeach. Let A i,L denote { L } ∗ A i , and similarly B L denote { L } ∗ B . Let C i,L be thesum of the chain L { L } ∗ C i and the homology P ( A i , B, L ) (constructed in theprevious section from ∂L { L } ∗ C i to [ { L } ∗ A i , { L } ∗ B ]). So C i,L has suplength . L , volume . L log L , and boundary [ { L } ∗ A i , { L } ∗ B ] = [ A i,L , B L ].Similarly, for each i take the homology P ( A i , A i , L ) constructed in the pre-vious section from L { L } [ A i , A i ] to [ { L } ∗ A i , { L } ∗ A i ]. Then define the 3-chain D L := L { L } ∗ D + X i =1 P ( A i , A i , L )10hich has suplength . L , volume . L log L , and boundary P i =1 [ { L } ∗ A i , { L } ∗ A i ] = P i =1 [ A i,L , A i,L ].Then the 4-cycle Z L is Z := [ B L , D L ] + 2 X i =1 [ A i,L , C i,L ] . The boundary of this is ∂Z L = [ B L , X i =1 [ A i,L , A i,L ]] + 2 X i =1 [ A i,L , [ B L , A i,L ]]which vanishes due to the Jacobi identity applied to A i,L , A i,L and B L .To prove Theorem A we will show the following bounds on the suplength of Z L , the volume of Z L , and the homological degree of Z L (with respect to some β ∈ H (Ω Y )). Proposition 4.1.
There exist constants
C, c > and a β ∈ H (Ω Y ) such thatthe -cycles Z L constructed above satisfy:1. Suplength ( Z L ) ≤ CL V ol ( Z L ) ≤ CL log L h β, Z L i > cL The proof of this occupies the rest of the paper. We start with the suplengthbound. By construction, the suplength of each chain A i,L , B L , C i,L and D L is L times the suplength of the corresponding chain without the subscript L .Suplength is additive under the bracket, so the suplength bound clearly holdswhen C is taken to be at least2 max { Suplength(Ψ) | Ψ ∈ { A i , B, C i , D }} . Similarly we obtain the volume bound by inspecting the volume of each of A i,L , B L , C i,L and D L . The cycles A i,L := { L } ∗ A i and B L := { L } B satisfyVol( A i,L ) ≤ Vol( A i ) and Vol( B L ) ≤ Vol( B ), as { L } is 1-Lipschitz on constantcurfew subspaces of Ω X so nonincreasing in the volume of constant curfewchains. The chain D L is the sum of L { L } ∗ D (volume ≤ L Vol( D ) and ahomology Y constructed in the previous section (volume ≤ CL log L ). HenceVol( D L ) ≤ CL log L for some C and similarly with the C i,L . The volume ofthe bracket is the product of the volumes (up to a constant factor) and so thevolume bound on Z L follows.The proof of the final part of the proposition occupies the next subsection.11 .2 Homological degree of Z L We will evaluate Z L against a cohomology class in H (Ω Y ) using Chen’s iteratedintegrals. These are differential forms defined on Ω Y , built out of differentialforms on the underlying space Y . For a rigorous setup of the smooth structureon Ω Y and discussion of the construction of iterated integrals, a good reviewis [Gug77]. See also original papers [Che73], [Che77] a concise but thoroughoverview in Hain’s thesis [Hai84], and also [Ell20] for a further discussion of theinterplay between iterated integrals on Ω X and a Riemannian metric on X .The third part of Proposition 4.1 is established via the following claim. Claim.
There is a nonzero cohomology class β ∈ H (Ω Y ) with the two followingproperties. First, the image of β under the dual of the real Hurewicz map H (Ω Y ) → Hom( π ( Y ) ⊗ R , R ) is nontrivial. Second, h β, Z L i = L h β, Z i 6 = 0.Here’s the β that we will use. The 2-cells ˜ A i and ˜ B in Y give a basisfor H ( Y ). Take differential forms a i , b on Y that represent the dual basisof H ( Y ), such that when restricted to the 2-skeleton Y (2) , each a i and b isonly supported on its respective 2-cell and their supports do not contain thebasepoint. Similarly, the 4-cells of Y are ˜ C i and ˜ D . These give a basis of H ( Y ). A dual basis of H ( Y ) is represented by c i := a i b and d := a (the a i are all cohomologous). Note that a i b vanishes identically on the 2-skeleton Y (2) of Y . Then consider the iterated integral ∫ a c , a closed differential 4-form onΩ Y . Then β is the cohomology class of this ∫ a c .Intuitively, this cohomology class can be thought of as follows. Fix a rel-ative 4-submanifold P D ( a ) in ( Y, ∂Y ) Poincar´e dual to a ˜ A , and relative2-submanifold P D ( c ) in ( Y, ∂Y ) Poincar´e dual to c . Given a 4-cycle Z inΩ Y . the pairing h∫ c a , Z i counts, with appropriate multiplicity, the numberof loops in the support of Z that first pass through P D ( c ) (a codimension 3condition) and then pass through P D ( a ) (a codimension 1 condition). I wouldlike to thank Dev Sinha for explaining this way of thinking to me.First we show that under the dual of the real Hurewicz, β is nontrivial inHom( π ( Y ) ⊗ R , R ). We do this by showing that the cycle Z is primitive (soits homology class in the image of the real Hurewicz map π (Ω Y ) → H (Ω Y ))and showing that h β, Z i 6 = 0.The cycle Z is primitive for algebro-topological reasons. The chains A i , B, C i , D generate the Adams-Hilton chain algebra AH ( Y ) which is a model for C ∗ (Ω Y ).Working rationally, the primitively generated Hopf algebra structure on AH ( Y )gives an isomorphism between H ∗ ( AH ( Y )) and H ∗ (Ω Y ) as Hopf algebras [Qui69][MM65]. The upshot of this is that any cycle constructed out of A i , B , C i , D and the Lie bracket on cycles is primitive.Next we will evaluate β our cycles Z and Z L . To do this we will employthe following two lemmas about evaluation of iterated integrals. Lemma 4.1. (Chen, [Che73]) Let M be a simply connected smooth manifoldwith finite Betti numbers. Let Z and Z be chains on Ω M and Z Z their roduct. If ω , ω , . . . , ω r are smooth forms on M then h∫ ω ω . . . ω r , Z Z i = r X i =0 h∫ ω . . . ω i , Z ih∫ ω i +1 . . . ω r , Z i . In both lemmas, the convention is that the paring vanishes if the dimensionof the differential form is not equal to the dimension of the chain that it is beingevaluated.
Lemma 4.2. (Chen, [Che77]) Let M be a simply connected smooth manifoldwith finite Betti numbers. Let Z be a n -chain on Ω M with suspension ˆ Z ∈ C n +1 ( M ) . Let ω be a differential form on M . Then h∫ w, Z i = h ω, ˆ Z i . Using these lemmas and the description of Z = [ B, D ] + 2 P i =1 [ A i , C i ] wecan compute: h∫ a c , Z i = h∫ a c , BD + DB + 2 X i =1 ( A i C i + C i A i ) i = h∫ a , B ih∫ c , D i + 2 X i =1 h∫ a , A i ih∫ c , C i i = 0 · X i =1 δ i · δ i = 2 = 0 . Next, we evaluate ∫ a c on Z L . Recall Z L = [ B L , D L ] + 2 P i =1 [ A i,L , C i,L ]. Lemma 4.3.
Let X be an arbitrary topological space and let Z be a spherical n -cycle in Ω X . Then on the level of homology, [ { L } ∗ Z ] = L [ Z ] .Proof. Let f : S n → Ω X be a map and ω n an n -cycle representing the fun-damental class such that f ∗ ω n = Z . Let Y n,L be an ( n + 1)-chain in ( S n ) × L with boundary Diag( ω n ) − Bouquet( ω n ). Then ( f × f × · · · × f ) ∗ Y n,L is a ho-mology between { L } ∗ Z and a cycle that, after a reparametrization of loops, ishomologous to LZ .Hence on the level of homology [ B L ] = L [ B ] and similarly [ A i,L ] = L [ A i ].Since the 1-form ∫ a i is closed on Ω X , this gives us that h∫ a i , B L i = 0 and h∫ a i , A i,L i = L · δ i . This gives h∫ a c , Z L i = 0 · h∫ c , D i + 2 L · h∫ c , C ,L i (1)It remains to compute h∫ c , C ,L i . The 3-chain C ,L is the sum of L { L } ∗ C and a 3-chain Y constructed in the proof of Theorem B. In particular, P issupported in the subspace Ω( Y (2) ) ⊂ Ω Y because the A i and B are supported13n Ω( Y (2) ) too. The differential 4-form c on Y vanishes on Y (2) , and hence ∫ c vanishes on Ω( Y (2) ). Thus h∫ c , C ,L i = h∫ c , L { L } ∗ C i = L h∫ c , { L } ∗ C i . We can evaluate h∫ c , { L } ∗ C i using Lemma 4.2. The suspension c C is the4-cell ˜ C and the suspension \ { L } ∗ C is L copies of ˜ C stuck together. For a 4-chain Z in Y with boundary lying in Y (2) , h c , Z i measures the degree (relativeto Y (2) over the 4-cell ˜ C i . Hence h c , \ { L } ∗ C i = L . Plugging this back into (1)gives h∫ a c , Z L i = 2 L = L h∫ a c , Z i as required. References [Sam53] Hans Samelson. “
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