An Example Using Improved Lefschetz Duality∗

Pascal LAMBRECHTSJeremy LANEDonald STANLEY

1 Introduction

Let f:N→M be an embedding of closed oriented manifolds of codimension k.Classical Lefschetz duality implies that there is a long exact sequence of H∗M modules

where f!is the unique map preserving orientations.This gives us a short exact sequence 0→ cokerf!→ H∗MN → kerf!→ 0.Over the rationals this determines H∗MN as a vector space,but knowing its H∗(M)modules structure requires solving an extension problem.

One of the ways to think of the improved Lefschetz duality of[4]is that the sequence(1.1)can be lifted to a triangle in the derived category of C∗M(really AplM)dgmodules

whose associated long exact sequence on homology is the sequence(1.1).Hereef!is the unique map in the derived category of C∗M modules that preserves orientation classes.Since any map in a triangle determines the triangle,the mapef!determines C∗(MN)as a C∗M dg-module and hence determines H∗(MN)as an H∗M module.

The point of the short note is to describe two embeddings

whose homological shriek maps are the same

but whose complements S2n×Smf(S4n−1)and S2n×Smg(S4n−1),have different H∗(S2n×Sm)module structures.This gives an example of how the improved Lefschetz duality sequence(1.2)gives more information than classical Lefschetz duality.The easy computation also provides another example of how keeping chain level information a little longer can be helpful.

We begin in Section 2 by recalling the definition of a Poincar´e embedding and theorems due to Klein and Wall which allow us to apply improved Lefschetz duality to maps S4n−1→S2n×Sm.In Section 3,we give the necessary definitions for algebraic models from rational homotopy theory,and in Section 4 we provide the details of our example.

2 Poincar´e Embeddings and Smooth Embeddings

The main example of this paper is concerned with applying improved Lefschetz duality to algebraic models of continuous maps

The theorem requires that the algebraic map is a model of a Poincar´e embedding.Conveniently,every such map is a Poincar´e embedding for sufficiently large m due to a result of Klein.For further details,the reader is referred to the survey[3].

Morally,an embedding of manifolds N→M is a decomposition of the codomain M=T ∪∂TC,where T and C are codimension 0 sub-manifolds with boundary∂T=T ∩C,and we think of T as a tubular neighborhood of N,or we can think of N as a manifold with boundary that is a codimension 0 submanifold of M.The topological generalization of a closed manifold(manifold with boundary)is a Poincar´e duality space(pair),which is a finite CW-complex(pair)that satisfies Poincar´e duality.This motivates the following definition(the map i is required to be(w−p−1)-connected for transversality reasons).

definition 2.1LetWandPbe finite CW-complexes.IfWis a connected Poincar´e duality space of dimensionw,andPhas dimensionp,then a Poincar´e embedding of codimensionw − pis a commutative diagram of topological spaces

such that(2.1)is a homotopy pushout;(P,∂T)and(C,∂T)are Poincare duality pairs in dimensionw;and the mapiis(w−p−1)-connected.Ifffi ts into such a commutative diagram then we say thatfPoincar´e embeds.

Theorem 2.1 (see[2])LetPbe a finite CW-complex of dimension≤pandWawdimensional connected Poincar´e duality space.If a mapf:P → Wisr-connected thenfPoincar´e embeds provided thatw ≥ p+3and

Instead of using the notion of Poincar´e embeddings,we may replace “Poincar´e embedding”in the statement of Theorem 4.1 with “smooth embedding” and apply the following theorem.

Theorem 2.2(see[5])IfPandWare smooth manifolds withp≤w−3,a mapf:P→WPoincar´e embeds,and2w ≥ 3(k+1),thenfis homotopic to a smooth embedding.

3 Algebraic Models in Rational Homotopy Theory

In order to discuss models of embeddings,we must introduce some notation from rational homotopy theory.Details can be found in[1].

definition 3.1A differential graded algebra(DGA for short)Ris a cochain complex(V,d)overQwith an associative multiplication,which is a morphism of cochain complexesV⊗V →Vand an identity element1∈V0such that

We say thatRis commutative,or a CDGA,ifab=(−1)|a||b|bafor alla,b∈ R.

definition 3.2Given a DGAR,a leftR-DG module is a cochain complexMwith a cochain morphism·:R⊗M → Mcommuting with the differential inMsuch that(xy)·m=x·(y·m)and1·m=m.

Example 3.1 If φ :M → N and ρ :R → M are morphisms of DGA’s,then we can consider φ as a morphism of left R-DG modules by letting r·m= ρ(r)m and r·n= φ◦ρ(r)n for m ∈ M and n∈N.Similarly,we can consider the dual M∗=Hom(M,Q)of a DGA as right R-DG modules by letting f ·r=f(r·−)for f ∈ M∗.The differential in the dual is defined by

Recall that if M is an R-DG module,then for an integer k,the differential on the suspension skM is given by

and in general the symbol skcommutes with other symbols following the Kozul sign convention as if it were an element or function of degree k.

definition 3.3The cone of a morphismφ :M → Nof leftR-DG modules is defined as the leftR-DG moduleC(φ)=N ⊕φsMwith differentiald(n,sm)=(dn+ φm,−sdm)andR-DG module structure

4 Complements of Poincar´e Embeddings S4n−1 → S2n × Sm

In this section we use improved Lefschetz duality(see Theorem 4.1)to construct an example(Example 4.1)of two maps that have the same shriek map,but give different extensions in the long exact sequence(1.1).

Theorem 4.1 (Improved Lefschetz Duality,see[4])Consider a Poincar´e embedding(2.1)withWconnected and of dimensionk.Suppose that a quasi-isomorphism of differential graded algebrasρ :A → APL(W)has been given and letφ :R → Qbe anA-DG module model off:P→W.Then there is an isomorphism

ofH∗(W;Q)-modules.

Theorem 4.2Fixn≥1andm ≥2n+1,m4n−1.Letf,g:S4n−1→ S2n×Smbe Poincar´e embeddings such thatfis homotopically trivial andgis rationally homotopy nontrivial.Then

asH∗(S2n×Sm)-module maps,butasH∗(S2n×Sm)-modules.

Proof For simplicity,we will write the dimension k:=2n+m in the proof.Consider the Sullivan algebras Q=(Λ(x),0)and R=(Λ(a,b,c),d)over Q with|x|=4n − 1,|a|=2n,|c|=m and db=a2.Then R is a model of S2n× Smand Q is a model of S4n−1,so every map S4n−1→ S2n×Smhas a model of the form R → Q.For degree reasons,a morphism φα:R → Q of CDGA’s is uniquely determined by φα(b)= αx for α ∈ Q.

By improved Lefschetz duality,H∗(s−kR∗⊕s−kφ∗s(s−kQ∗))is isomorphic to the cohomology of the complement of f as described in definition 2.1 as an H∗(S2n×Sm;Q)-module.We denote the cone of the suspended dual map s−kφ∗α:s−kQ∗→ s−kR∗by

To compute Cα,let R and Q have bases

respectively,and denote an element of the dual basis with an asterisk(for example,a∗(a)=1).For any α∈Q,one can show that

The module structure on H∗(Cα)is independent of α,except in degree m − 2n,where the kernel of d depends on φα.Consider a homogeneous element

in Cαof degree m − 2n with βi∈ Q.Observe that

which is 0 if and only if β1=αβ2.Hence,the cocycles inare of the form

and are coboundaries if and only if β=0.The R-DG module action on a representative of such a cohomology class is

Hence,in H∗(Cα)

However,the induced map on H∗is trivial for every map S4n−1→S2n×Smfor degree reasons,so it is clear that f!=g!.Furthermore,if α0 then the following sequence of H∗(A)modules does not split at H∗(Cα)(in other words,H∗(Cα)is not a trivial extension).

Example 4.1 For all n≥1 and m≥2n+1,m4n−1,there are smooth embeddings

such that

as H∗(S2n×Sm)-module maps,but

as H∗(S2n×Sm)-modules.

Proof Let

be a trivial map and

be the composition of the self-Whitehead product with the inclusion.By Theorem 2.2 there exist embeddings f ≃ f′and g ≃ g′.Since g′and hence g are rationally nontrivial the example follows directly from Theorem 4.2.

[1]F´elix,Y.,Halperin,S.and Thomas,J.C.,Rational homotopy theory,Graduate Texts in Mathematics,205,Springer-Verlag,New York,2001.

[2]Klein,J.R.,Poincar´e duality embeddings and fi berwise homotopy theory,Topology,38(3),1999,597–620.

[3]Klein,J.R.,Poincar´e duality spaces,Surveys on Surgery Theory,Vol.1,Ann.of Math.Stud.,Vol.145,Princeton Univ.Press,Princeton,NJ,2000,135–165.

[4]Lambrechts,P.and Stanley,D.,Algebraic models of Poincar´e embeddings,Algebr.Geom.Topol.,5,2005,135–182(electronic).

[5]Wall,C.T.C.,Classification problems in differential topology.VI.Classification of(s − 1)-connected(2s+1)-manifolds,Topology,6,1967,273–296.