A controlled local-global theorem for simplicial complexes
aa r X i v : . [ m a t h . A T ] N ov A CONTROLLED LOCAL-GLOBAL THEOREM FOR SIMPLICIALCOMPLEXES
SPIROS ADAMS-FLOROUA
BSTRACT . In this paper we prove that a simplicial map of finite-dimensional locallyfinite simplicial complexes has contractible point inverses if and only if it is an ǫ -controlled homotopy equivalence for all ǫ > if and only if f × id R is a boundedhomotopy equivalence measured in the open cone over the target. This confirms forsuch a space X the slogan that arbitrarily fine control over X corresponds to boundedcontrol over the open cone O ( X + ) . For the proof a one parameter family of cellulations { X ′ ǫ } <ǫ<ǫ ( X ) is constructed which provides a retracting map for X which can be usedto compensate for sufficiently small control.
1. I
NTRODUCTION
A homeomorphism has point inverses which are all points. If a map f is homotopicto a homeomorphism it is reasonable to suppose that f might have point inverses thatare ‘close’ to being points in some suitable sense. Controlled topology takes ‘close’ tomean small with respect to a metric. One then studies maps with small point inversesand attempts to prove that such a map is homotopic to a homeomorphism.This approach has many successes in the literature: as a consequence of Chapmanand Ferry’s α -approximation theorem ([CF79]) a map between closed metric topolog-ical manifolds with sufficiently small point inverses is homotopic to a homeomorph-ism through maps with small point inverses. One can also consider maps where thepoint inverses all have the homotopy groups of a point, i.e. are contractible. In thenon-manifold case Cohen proves in [Coh67] that a p.l. map of finite polyhedra withcontractible point inverses is a simple homotopy equivalence.When doing controlled topology it is desirable that the space we consider, X , comesequipped with a metric. In the absence of a metric it is sufficient that X has at least amap p : X → M to a metric space ( M, d ) , called a control map , which then allows us tomeasure distances in M . In general to be able to detect information about X from thecontrol map and the metric on M we would ideally like p to be highly connected.Let f : X → Y be a map of spaces equipped with control maps p : X → M , q : Y → M to a metric space ( M, d ) . We say that f : ( X, p ) → ( Y, q ) is ǫ -controlled if f commutes with the control maps p and q up to a discrepancy of ǫ , i.e. for all x ∈ X , d ( p ( x ) , qf ( x )) < ǫ . We say that f : ( X, p ) → ( Y, q ) is an ǫ -controlled homotopy equivalence if there exists a homotopy inverse g and homotopies h : g ◦ f ∼ id X and h : f ◦ g ∼ id Y such that all of f : ( X, p ) → ( Y, q ) , g : ( Y, q ) → ( X, p ) , h : ( X × R , p × id R ) → ( X, p ) and h : ( Y × R , q × id R ) → ( Y, q ) are ǫ -controlled maps.Note that an ǫ -controlled homotopy equivalence f and its inverse g do not movepoints more than a distance ǫ when measured in M and that the homotopy tracks areno longer than ǫ when measured in M . If X and Y are also metric spaces it is perfectlypossible that the homotopy tracks are large in X or Y and only become small aftermapping to M . Controlled topology is not functorial because the composition of two maps withcontrol less than ǫ is a map with control less than ǫ . This motivates Pedersen’s de-velopment in [Ped84b] and [Ped84a] of bounded topology, where the emphasis is nolonger on how small the control is but rather just that it is finite. A map f : ( X, p ) → ( Y, q ) is called bounded if f commutes with the control maps up to a finite discrepancy B . Similarly a bounded homotopy equivalence is one where all the maps and homo-topies are bounded. Bounded topology is functorial as the sum of two finite boundsremains finite.In [FP95] Ferry and Pedersen suggest a relationship between controlled topologyon a space X and bounded topology on the open cone O ( X + ) when they write in afootnote:“It is easy to see that if Z is a Poincar´e duality space with a map Z → K such that Z has ǫ -Poincar´e duality for all ǫ > when measured in K (after subdivision), e.g. a homology manifold, then Z × R is an O ( K + ) -bounded Poincar´e complex. The converse (while true) will not concernus here.”For X a proper subset of S n the open cone O ( X + ) ⊂ R n +1 is the union of all rays fromthe origin ∈ R n +1 through points in X + = X ⊔ { x } together with the subspacemetric. There is a natural map j X : X × R → O ( X + ) , called the coning map , given by j X ( x, t ) := (cid:26) tx, t > , − tx , t . See section 2 for a more general definition of the open cone and the coning map formore general metric spaces.The footnote above leads one to conjecture that f : ( X, qf ) → ( Y, q ) is an ǫ -controlledhomotopy equivalence for all ǫ > if and only if f × id R : ( X × R , j Y ( qf × id R )) → ( Y × R , j Y ) is a bounded homotopy equivalence. In this paper we prove this conjecture for thecase of a simplicial map of finite-dimensional locally finite (henceforth f.d. l.f.) sim-plicial complexes measured in the target. We may measure in the target since suchcomplexes come naturally equipped with a path metric. We prove Theorem 1.
Let f : X → Y be a simplicial map of f.d. l.f. simplicial complexes with Y equipped with the path metric. Then the following are equivalent:(i) f has contractible point inverses,(ii) f : ( X, f ) → ( Y, id Y ) is an ǫ -controlled homotopy equivalence for all ǫ > ,(iii) f × id R : ( X × R , j Y ( f × id R )) → ( Y × R , j Y ) is a bounded homotopy equivalence.Working with simplicial maps makes life much easier - one needs only check thatthe point inverses of the barycentres are contractible: Proposition 2.
Let f : X → Y be a simplicial map of l.f. f.d. simplicial complexes.Then(i) for all simplices σ ∈ Y , there is a p.l. isomorphism f − (˚ σ ) ∼ = f − ( b σ ) × ˚ σ, (ii) f has contractible point inverses if and only if f − ( b σ ) is contractible for all σ ∈ Y .Moreover, simplicial maps allow us to ‘lift’ certain properties of the target spaceto the preimage, in particular the fact that open stars deformation retract onto opensimplices: CONTROLLED LOCAL-GLOBAL THEOREM FOR SIMPLICIAL COMPLEXES 3
Proposition 3.
Let f : X → Y be a simplicial map of f.d. l.f. simplicial complexes.Then for all σ ∈ Y , f − (st( σ )) p.l. deformation retracts onto f − (˚ σ ) . If, as in the theorem, we additionally suppose that a simplicial map f : X → Y hascontractible point inverses, then f turns out to have the approximate homotopy liftingproperty : for all ǫ > , the lifting problem Z × { } h / / (cid:15) (cid:15) X f (cid:15) (cid:15) Z × I e H ǫ ; ; ✇✇✇✇✇ H / / Y has a solution e H ǫ : Z × I → X such that the diagram commutes up to ǫ , i.e. d Y ( H ( z, t ) , f ( e H ǫ ( z, t )) < ǫ for all ( z, t ) ∈ Z × I , where d Y is the metric on Y . This is precisely the definition of anapproximate fibration given by Coram and Duvall in [CD77].The key ingredient in proving ( i ) ⇒ ( iii ) and obtaining the approximate homo-topy lifting property is the construction and use of the fundamental ǫ -subdivision cellu-lation X ′ ǫ of an f.d. l.f. simplicial complex X . The X ′ ǫ are a family of cellulations with lim ǫ → X ′ ǫ = X similar to the family of cellulations obtained by taking slices || X || × { t } of the prism || X || × [0 , triangulated so that || X || × { } is given the triangulation X and || X || × { } the barycentric subdivision Sd X . The key difference is that the cel-lulations X ′ ǫ are defined in such a way as to guarantee that the homotopy from X ′ ǫ to X through X ′ δ for δ ∈ (0 , ǫ ) has control ǫ . These cellulations provide retracting mapsthat compensate for ǫ -control when proving squeezing results. This is precisely whatis missing when trying to prove such results for a more general class of spaces.Section 2 recaps some necessary preliminaries. In section 3 the fundamental ǫ -subdivision cellulation of an f.d. l.f. simplicial complex is defined and a few usefulproperties explained. In section 4 Propositions 2 and 3 are proved and consequently adirect proof of Theorem 1 is given. Acknowledgement.
This work is partially supported by Prof. Michael Weiss’ Hum-boldt Professorship. 2. P
RELIMINARIES
In this paper only locally finite finite-dimensional simplicial complexes will be con-sidered. Such a space X shall be given a metric d X , called the standard metric , asfollows. First define the standard n -simplex ∆ n in R n +1 as the join of the points e =(1 , , . . . , , . . . , e n = (0 , . . . , , ∈ R n +1 . ∆ n is given the subspace metric d ∆ n of thestandard ℓ -metric on R n +1 . The locally finite finite-dimensional simplicial complex X is then given the path metric whose restriction to each n -simplex is d ∆ n . Distancesbetween points in different connected components are thus ∞ . See § of [Bar03] orDefinition . of [HR95] for more details.Let p : Y → X be a simplicial map of locally-finite simplicial complexes equippedwith standard metrics. For σ a simplex in Y , the diameter of σ measured in X is diam( σ ) := sup x,y ∈ σ d X ( p ( x ) , p ( y )) . By a deformation retract we mean a strong deformation retract.
SPIROS ADAMS-FLOROU
The radius of σ measured in X is rad( σ ) := inf x ∈ ∂σ d X ( p ( b σ ) , p ( x )) . The mesh of X measured in Y is mesh( X ) := sup σ ∈ X { diam( σ ) } . The comesh of X measured in Y is comesh( X ) := inf σ ∈ X, | σ |6 =0 { rad( σ ) } . Using the standard metric on X and id X : X → X as the control map diam( σ ) = √ and rad( σ ) = √ | σ | ( | σ | +1) , for all σ ∈ X , so consequently mesh( X ) = √ and if X is n -dimensional comesh( X ) = √ n ( n +1) .The open star st ( σ ) of a simplex σ ∈ X is defined by st( σ ) := [ τ > σ ˚ τ . The open cone was first considered by Pedersen and Weibel in [PW89] where itwas defined for subsets of S n . This definition was extended to more general spacesby Anderson and Munkholm in [AM90]. We make the following definition: For acomplete metric space ( M, d ) the open cone O ( M + ) is defined to be the identificationspace M × R / ∼ with ( m, t ) ∼ ( m ′ , t ) for all m, m ′ ∈ M if t . We define a metric d O ( M + ) on O ( M + ) by setting d O ( M + ) (( m, t ) , ( m ′ , t )) = (cid:26) td ( m, m ′ ) , t > , , t ,d O ( M + ) (( m, t ) , ( m, s )) = | t − s | and defining d O ( M + ) (( m, t ) , ( m ′ , s )) to be the infimum over all paths from ( m, t ) to ( m ′ , s ) , which are piecewise geodesics in either M × { r } or { n } × R , of the length ofthe path. I.e. d O ( M + ) (( m, t ) , ( m ′ , s )) = max { min { t, s } , } d X ( m, m ′ ) + | t − s | . This metric is carefully chosen so that d O ( M + ) | M ×{ t } = (cid:26) td O ( M + ) | M ×{ } , t > , , t . This is precisely the metric used by Anderson and Munkholm in [AM90] and alsoby Siebenmann and Sullivan in [SS79], but there is a notable distinction: we do notnecessarily require that our metric space ( M, d ) has a finite bound.There is a natural map j X : X × R → O ( X + ) given by the quotient map X × R → X × R / ∼ ( x, t ) [( x, t )] . We call this the coning map .For M a proper subset of S n with the subspace metric, the open cone O ( M + ) can bethought of as the subset of R n +1 consisting of all the points in the rays out of the originthrough points in M + := M ∪ { pt } with the subspace metric. This is not the same asthe metric we just defined above but it is Lipschitz equivalent. CONTROLLED LOCAL-GLOBAL THEOREM FOR SIMPLICIAL COMPLEXES 5
3. S
UBDIVISION CELLULATIONS
In this section we construct a controlled -parameter family of subdivision cellu-lations of X which shall be used later in constructing controlled homotopies. This -parameter family is defined in analogy to the -parameter family of subdivisioncellulations obtained by restricting a triangulation of the prism X × I to the slices { X × { t }} The canonical triangulation of || X || × I from X to Sd X is defined to haveone ( | σ | + n + 1) -simplex ( σ × { } ) ∗ ( b σ . . . b σ n × { } ) in || X || × I for every chain of inclusions in X of the form σ σ < . . . < σ n . With a slight abuse of terminology we shall call such a chain of inclusions a flag in X of length n . It may easily be verified that this indeed gives a triangulation. Example 3.2. Let X be a -simplex. Figure 1 illustrates the canonical triangulation of || X || × I and what the induced cellulations of the slice || X || × { . } is.PSfrag replacements t = 0 t = 0 . t = 1 F IGURE 1. Obtaining cellulations from the prism.The slices {|| X || × { t }} Define the flag cellulation of X by χ ( X ) := dim( X ) [ m =0 [ σ σ <...<σ m σ × b σ . . . b σ m ⊂ X × Sd X. SPIROS ADAMS-FLOROU Observe that χ ( X ) has the same cellulation as that inherited by || X || × { t } for any t ∈ (0 , from the canonical triangulation of the prism from X to Sd X . We nowconstruct a -parameter family of p.l. isomorphisms Γ ǫ : χ ( X ) → X which shall beused to give X a -parameter family of cellulations. Definition 3.4. For ǫ < comesh( X ) define a map Γ ǫ : χ ( X ) → X by Γ ǫ ( v × b v ) := v, for all vertice v ∈ X, Γ ǫ ( v × b τ ) := ∂B ǫ ( v ) ∩ b v b τ , for all inclusions of a vertex v < τ, where ∂B ǫ ( v ) is the sphere of radius ǫ centred at the vertex v and ∂B ( v ) := v .Extend Γ ǫ piecewise linearly over each cell of χ by Γ ǫ : σ × b σ . . . b σ m → X ( s , . . . , s n , t , . . . , t m ) n X i =0 m X j =0 s i t j Γ ǫ ( v i × b σ j ) , where σ = v . . . v n with barycentric coordinates ( s , . . . , s n ) and b σ . . . b σ m has barycen-tric coordinates ( t , . . . , t m ) .We call the image under Γ ǫ of the flag cellulation the fundamental ǫ -subdivision cellu-lation of X and denote it by X ′ ǫ . We use the following notation for the cells of X ′ ǫ : Γ σ ,...,σ m ( σ ) := Γ ǫ ( σ × b σ . . . b σ m ) , Γ σ ,...,σ m (˚ σ ) := Γ ǫ (˚ σ × b σ . . . b σ m ) , for all flags σ σ < . . . < σ n . (cid:3) Example 3.5. Let X be the simplex σ = v v v with faces labelled τ = v v , τ = v v and τ = v v , then the fundamental ǫ -subdivision cellulation of X is as in Figure 2. Each Γ σ ,...,σ i ( τ ) is the closed cell pointed to by the arrow. (cid:3) PSfrag replacements ρ ρ ρ ǫ Γ τ ,σ ( τ )Γ τ ( τ ) Γ σ ( τ )Γ ρ ,τ ,σ ( ρ ) Γ ρ ,σ ( ρ )Γ τ ,σ ( ρ ) Γ ρ ,τ ( ρ )Γ ρ ,σ ( ρ ) Γ σ ( ρ ) Γ ρ ( ρ )Γ σ ( σ ) F IGURE 2. The cellulation X ′ ǫ for a -simplex. CONTROLLED LOCAL-GLOBAL THEOREM FOR SIMPLICIAL COMPLEXES 7 Remark 3.6. Note that for all < ǫ < comesh( X ) , Γ ǫ is a p.l. isomorphism and that Γ = pr : X × Sd X → X . Hence Γ δ ◦ Γ − ǫ : X ′ ǫ → X ′ δ is a p.l. isomorphism for all < ǫ, δ < comesh( X ) . Further, for < ǫ < comesh( X ) thecellulation X ′ ǫ is homotopic to X via the straight line homotopy h ,ǫ : Y × I → Y ( y, t ) Γ ǫ (1 − t ) Γ − ǫ ( y ) . This homotopy sends each vertex Γ τ ( v ) to the point v along a straight line of length precisely ǫ .Convexity of the cells of Y ′ ǫ guarantees that all homotopy tracks are of length at most ǫ . Hence h ,ǫ has control ǫ . 4. P ROOF OF MAIN THEOREM In this section we prove the main theorem which we restate for convenience. Theorem 1. Let f : X → Y be a simplicial map of f.d. l.f. simplicial complexesequipped with their path metrics and let j Y : Y × R → O ( Y + ) be the coning map.Then the following are equivalent:(i) f has contractible point inverses,(ii) f : ( X, f ) → ( Y, id Y ) is an ǫ -controlled homotopy equivalence for all ǫ > ,(iii) f × id R : ( X × R , j Y ( f × id R )) → ( Y × R , j Y ) is a bounded homotopy equivalence.To facilitate the proof of the main theorem we first require two propositions. Proposition 2. Let f : X → Y be a simplicial map of l.f. f.d. simplicial complexes.Then(i) for all simplices σ ∈ Y , there is a p.l. isomorphism f − (˚ σ ) ∼ = f − ( b σ ) × ˚ σ, (ii) f has contractible point inverses if and only if f − ( b σ ) is contractible for all σ ∈ Y . Proof. ( i ) : If ˚ σ is not in the image of f then the result holds as f − (˚ σ ) = f − ( b σ ) = ∅ . Let σ = w . . . w m be some simplex in Y . Suppose there is a τ ∈ X such that f ( τ ) = σ . Let f τ := f | τ : τ → σ . Since σ is the join of its vertices we have that τ = m ∗ i =0 f − τ ( w i ) with f − τ ( x ) ∼ = Q mi =0 f − τ ( w i ) ∼ = f − τ ( b σ ) for all x ∈ ˚ σ . Whence f − τ (˚ σ ) ∼ = f − τ ( b σ ) × ˚ σ. Suppose τ < τ are such that f ( τ i ) = σ for i = 0 , . Then f − τ (˚ σ ) ⊂ ∼ = f − τ (˚ σ ) ∼ = f − τ ( b σ ) × ˚ σ ⊂ f − τ ( b σ ) × ˚ σ. Thus we can reconstruct f − ( b σ ) from { f − τ ( b σ ) | f ( τ ) = σ } as f − ( b σ ) = [ τ : f ( τ )= σ f − τ ( b σ ) and consquently f − (˚ σ ) = [ τ : f ( τ )= σ f − τ (˚ σ ) ∼ = [ τ : f ( τ )= σ f − τ ( b σ ) × ˚ σ = f − ( b σ ) × ˚ σ. ( ii ) : Clear from the fact that f − ( x ) ∼ = f − ( b σ ) for x ∈ ˚ σ . (cid:3) SPIROS ADAMS-FLOROU This proposition tells us that for a simplicial map f with contractible point inverses,the restriction over each simplex, f | : f − (˚ σ ) → ˚ σ , is a trivial fibre bundle with fibre f − ( b σ ) ≃ ∗ . We will see that we can define a section over each simplex interior andthe contractibility of each f − ( b σ ) allows us to piece these local sections together byhomotopies that are large in X but can be made arbitrarily small in Y . This yields aglobal homotopy inverse g ǫ , for all ǫ > , that is an approximate section in the sensethat f ◦ g ǫ ≃ id Y via homotopy tracks of diameter < ǫ . This approximate sectioncan be used to approximately lift homotopies, hence we see that f is an approximatefibration. Proposition 3. Let f : X → Y be a simplicial map of f.d. l.f. simplicial complexes.Then for all σ ∈ Y , f − (st( σ )) p.l. deformation retracts onto f − (˚ σ ) . Proof. If f − (˚ σ ) is empty then so is f − (˚ ρ ) for all ρ > σ and hence f − ( st ( σ )) is emptyso the result holds vacuously.Suppose instead that f − (˚ σ ) ∼ = f − ( b σ ) × ˚ σ is non-empty. For every ρ > σ let σ Cρ ∈ Y be the unique simplex such that ρ = σ ∗ σ Cρ . For all τ ∈ X with f ( τ ) = ρ let f τ := f | τ : τ → ρ so that τ = f − τ ( σ ) ∗ f − τ ( σ Cρ ) . Every x ∈ ˚ τ ∪ f − τ (˚ σ ) can be written uniquelyas x = (1 − t ) x σ + tx σ Cτ , for x σ ∈ f − τ (˚ σ ) , x σ Cρ ∈ f − τ (˚ σ Cρ ) , t ∈ [0 , . Thus letting the t parameter go to at unit speed and staying there thereafter defines a linear (strong)deformation retraction of ˚ τ ∪ f − τ (˚ σ ) onto f − τ (˚ σ ) . The deformation retractions definedlike this for different simplices surjecting onto ρ agree on intersections and so glue togive a p.l. deformation retraction of f − (˚ ρ ∪ ˚ σ ) onto f − (˚ σ ) . These glue together togive the desired deformation retraction of f − ( st ( σ )) onto f − (˚ σ ) . (cid:3) Proof of Theorem 1. ( i ) ⇒ ( iii ) : Let f : X → Y be a simplicial map of f.d. l.f. sim-plicial complexes with contractible point inverses. Then f is necessarily surjective ascontractible point inverses are non-empty. We seek to define a one parameter familyof homotopy inverses { g ǫ : Y → X } <ǫ< comesh( Y ) and homotopies { h ,ǫ : id X ≃ g ǫ ◦ f } <ǫ< comesh( Y ) , { h ,ǫ : id Y ≃ f ◦ g ǫ } <ǫ< comesh( Y ) parametrised by control. Given such families we obtain a bounded homotopy inverse g to f × id R : ( X × R , j Y ◦ ( f × id R )) → ( Y × R , j Y ) defined by g : Y × R → X × R ;( y, t ) g α ( t ) ( y ) and bounded homotopies h : id X × R ≃ g ◦ ( f × id R ) : X × R × I → X × R ;( x, t, s ) h ,α ( t ) ( x, s ) ,h : id Y × R ≃ ( f × id R ) ◦ g : Y × R × I → Y × R ;( y, t, s ) h ,α ( t ) ( y, s ) , where α : R → (0 , comesh( Y )] is the function α : t (cid:26) comesh( Y ) , t / comesh( Y ) , /t, t > / comesh( Y ) . CONTROLLED LOCAL-GLOBAL THEOREM FOR SIMPLICIAL COMPLEXES 9 Give Y the fundamental ǫ -subdivision cellulation Y ′ ǫ as defined in Definition 3.4. Wedefine g ǫ , h ,ǫ and h ,ǫ by induction. First, define a map γ : χ ( Y ) → X by induction onthe flag length of cells in χ ( Y ) . Let γ b σ × ˚ σ : b σ → f − ( b σ ) be any map, then define γ on b σ × σ as the closure of the map γ b σ × ˚ σ × id ˚ σ : b σ × ˚ σ → f − ( b σ ) × ˚ σ ∼ = f − (˚ σ ) . Let Φ τ,σ : f − ( b σ ) → f − ( b τ ) denote the maps obtained in the closure of γ b σ × ˚ σ for τ < σ such that γ b σ × ˚ τ = (Φ τ,σ ◦ γ b σ × ˚ σ ) × id ˚ τ : b σ × ˚ τ → f − ( b τ ) × ˚ τ ∼ = f − (˚ τ ) . Now suppose that we have continuously defined γ on all cells of χ ( Y ) of flag lengthat most n and that the map takes the form γ b σ ... b σ i × ˚ σ × id ˚ σ : b σ . . . b σ i × ˚ σ → f − (˚ σ ) × ˚ σ ∼ = f − (˚ σ ) on each cell for i n for some maps γ b σ ... b σ i × ˚ σ : b σ . . . b σ i → f − (˚ σ ) . These maps define a map γ ∂ ( b σ ... b σ n +1 ) × ˚ σ : ∂ ( b σ . . . b σ n +1 ) → f − ( b σ ) which extends to a map γ b σ ... b σ n +1 × ˚ σ : b σ . . . b σ n +1 → f − ( b σ ) by the contractibility of f − ( b σ ) . Define γ on the cell b σ . . . b σ n +1 × σ as the closure ofthe map γ b σ ... b σ n +1 × ˚ σ × id ˚ σ : γ b σ ... b σ n +1 × ˚ σ → f − ( b σ ) × ˚ σ ∼ = f − (˚ σ ) . By induction this defines the map γ .For all < ǫ < comesh( Y ) , set g ǫ := γ ◦ Γ − ǫ : Y → χ ( Y ) → X. We claim that { g ǫ } <ǫ< comesh( Y ) is a one parameter family of homotopy inverses to f parametrised by control.Consider first the composition f ◦ g ǫ . f ◦ γ = pr ◦ ( γ b σ ... b σ n × ˚ σ × id ˚ σ ) = pr : b σ . . . b σ n × ˚ σ → f − ( b σ ) × ˚ σ → ˚ σ . Hence f ◦ γ : χ ( Y ) ⊂ Sd Y × Y → Y is just projection onto Y , i.e. the map Γ =lim ǫ → Γ ǫ . Thus f ◦ g ǫ = ( f ◦ γ ) ◦ Γ − ǫ = Γ ◦ Γ − ǫ . Choosing h ,ǫ precisely as in Remark3.6 we have h ,ǫ : id X = Γ ǫ ◦ Γ − ǫ ≃ Γ ◦ Γ − ǫ = f ◦ g ǫ is an ǫ -controlled homotopy andin fact { h ,ǫ } <ǫ< comesh( Y ) is a one parameter of homotopies parametrised by control.Now consider the other composition: g ǫ ◦ f = γ ◦ Γ − ǫ ◦ f . Define a homotopy h ′ ,ǫ : X × I → X by h ′ ,ǫ = id f − ( b σ ) × h ,ǫ : f − ( b σ ) × ˚ σ × [0 , → f − (˚ σ ) with h ′ ,ǫ ( − , 1) := lim t → h ′ ,ǫ ( − , t ) . This homotopy is sent by f to h ,ǫ : f ( h ′ ,ǫ ( x, t )) = h ,ǫ ( f ( x ) , t ) , ∀ ( x, t ) ∈ X × I. Hence h ′ ,ǫ has control ǫ .We now seek a homotopy h ′′ ,ǫ : h ′ ,ǫ ( − , ≃ g ǫ ◦ f with zero control. Looking at f − (Γ ǫ ( ρ × ˚ σ )) for ρ = b σ . . . b σ n observe that h ′ ,ǫ ( − , is the closure of the map Φ σ ,σ n × h ,ǫ ( − , 1) = Φ σ ,σ n × (Γ ◦ Γ − ǫ ) : f − ( b σ n ) × Γ ǫ (˚ ρ × ˚ σ ) → f − ( b σ ) × ˚ σ , whereas g ǫ ◦ f is the closure of the map ( γ b σ ... b σ n × ˚ σ × id ˚ σ ) ◦ Γ − ǫ ◦ pr : f − ( b σ n ) × Γ ǫ (˚ ρ × ˚ σ ) → f − ( b σ ) × ˚ σ . The component of this map from Γ ǫ (˚ ρ ) to ˚ σ is Γ Γ − ǫ and so agrees with the com-ponent of h ′ ,ǫ ( − , to ˚ σ . We now find inductively a homotopy h ′′ ,ǫ : h ′ ,ǫ ( − , ≃ g ǫ ◦ f which only moves things in the fibre direction and hence has control. This isachieved precisely as before using the contractibility of the fibres. The concatenation h ,ǫ := h ′′ ,ǫ ∗ h ′ ,ǫ is an ǫ -controlled homotopy id X ≃ g ǫ ◦ f . As we use the same ho-motopies in the fibre direction for all < ǫ < comesh( Y ) this gives a one parameterfamily { h ,ǫ : id Y ≃ g ǫ ◦ f } <ǫ< comesh( Y ) parametrised by control as required.Note also that f | : f − ( τ ) → τ is a homotopy equivalence for all τ ∈ Y by restricting g ǫ , h ,ǫ and h ,ǫ . We call such a homotopy equivalence a Y -triangular homotopy equiva-lence . It is an open conjecture that f : X → Y is homotopic to a Y -triangular homotopyequivalence if and only if f is homotopic to an ǫ -controlled homotopy equivalence forall ǫ > . Y -triangular homotopy equivalences are discussed in [Ada13]. ( iii ) ⇒ ( ii ) : Let f × id have homotopy inverse g and homotopies h : id X × R ≃ g ◦ ( f × id R ) and h : id Y × R ≃ ( f × id R ) ◦ g all with bound at most B < ∞ . Let p t : R → { t } be projection onto t ∈ R .Let g t := (id X × p t ) ◦ g | Y ×{ t } : Y × { t } → X × R → X × { t } . This is a homotopyinverse to f × id { t } : X × { t } → Y × { t } with homotopies (id X × p t ) ◦ h | X ×{ t } : (id X × p t ) ◦ id X ×{ t } = id X ×{ t } ≃ (id X × p t ) ◦ ( g ◦ ( f × id R )) | X ×{ t } = (id X × p t ) ◦ g | Y ×{ t } ◦ ( f × id { t } )= g t ◦ ( f × id { t } ) and (id Y × p t ) ◦ h | Y ×{ t } : (id Y × p t ) ◦ id Y ×{ t } = id Y ×{ t } ≃ (id Y × p t ) ◦ (( f × id R ) ◦ g ) | Y ×{ t } = (id Y × p t ) ◦ ( f × id R ) ◦ g | Y ×{ t } = ( f × id { t } ) ◦ (id X × p t ) ◦ g | Y ×{ t } = ( f × id { t } ) ◦ g t . These homotopies have bound approximately B measured in Y × { t } ⊂ O ( Y + ) . Theslice Y × { t } has a metric t times bigger than Y = Y × { } , so measuring this in Y gives a homotopy equivalence f : X → Y with control proportional to Bt as required. ( ii ) ⇒ ( i ) : First note that a simplicial map f that is an ǫ -controlled homotopy equiv-alence for all ǫ > must be surjective. Suppose it is not, then there is a y ∈ Y \ im( f ) .Since f is simplicial ˚ σ ⊂ Y \ im( f ) where σ is the unique simplex of Y with y ∈ ˚ σ .Again since f is simplicial, if τ > σ we must have ˚ τ ⊂ Y \ im( f ) as well. Thus st ( σ ) = [ τ > σ ˚ τ ⊂ Y \ im( f ) . In particular the open star st ( σ ) is an open neighbourhood of y in Y \ im( f ) so we mayfind a ball B ǫ ′ ( y ) ⊂ Y \ im( f ) . Thus f cannot be an ǫ -controlled homotopy equivalence CONTROLLED LOCAL-GLOBAL THEOREM FOR SIMPLICIAL COMPLEXES 11 for ǫ < ǫ ′ as the homotopy tracks for the point y must travel a distance of at least ǫ ′ .This is a contradiction and so f is surjective.Each point y ∈ Y is contained in a unique simplex interior and hence in that sim-plex’s open star: ˚ σ ⊂ st( σ ) . Since the star is open there is an ǫ ′ such that B ǫ ′ ( y ) ⊂ st( σ ) .By hypothesis we can find an ǫ ′ -controlled homotopy inverse, g ǫ ′ , to f . Thus f − ( y ) is homotopic to g ǫ ′ ( y ) within f − (st( σ )) . By Proposition 3, f − (st( σ )) deformation re-tracts onto f − (˚ σ ) . By Proposition 2 this is p.l. isomorphic to f − ( b σ ) × ˚ σ which in turndeformation retracts onto f − ( b σ ) × { y } = f − ( y ) . Applying these two deformationretractions to the homotopy f − ( y ) ≃ g ǫ ′ ( y ) gives a contraction of f − ( y ) . Hence f has contractible point inverses. (cid:3) We conclude with an example illustrating the construction in the proof of ( i ) ⇒ ( iii ) . Example 4.1. Let , , , e = (1 , , , e = (0 , , and e = (0 , , be points in R . Define Y to be the simplicial complex with the following -simplices: σ := 0 ∗ e ∗ ( e + e ) and σ := 0 ∗ e ∗ ( e + e ) . Define X to be the simplicial complex with the following -simplices: τ := 0 ∗ e ∗ ( e + e ) , τ := e ∗ ( e + e ) ∗ ( e + e + e ) , τ := 0 ∗ e ∗ ( e + e + e ) and τ := 0 ∗ ( e + e ) ∗ ( e + e + e ) . The projection map f : X → Y ; ( x, y, z ) ( x, y, issimplicial and has contractible point inverses. Give Y the cellulation Y ′ ǫ for some small ǫ > as pictured in Figure 3.PSfrag replacements xy F IGURE ǫ -subdivision cellulations. We define g ǫ as in the proof by first defining maps γ b ρ × ˚ ρ : b ρ → f − ( b ρ ) for all ρ ∈ Y . Wedefine γ b ρ × ˚ ρ = , ˚ ρ ⊂ σ \ σ , / , ˚ ρ ⊂ σ ∩ σ , , ˚ ρ ⊂ σ \ σ . Then, for all ρ ∈ σ ∩ σ we choose the maps γ b ρ b σ i × ˚ ρ : b ρ b σ i → f − ( b ρ ) for i = 1 , as follows: γ b ρ b σ × ˚ ρ ( t , t ) = 12 t ,γ b ρ b σ × ˚ ρ ( t , t ) = 12 t + t where ( t , t ) are barycentric coordinates. Finally for v either vertex of σ ∩ σ and ρ the -simplex of σ ∩ σ we define the maps γ b v b ρ b σ i × ˚ v : b v b ρ b σ i → f − ( b v ) for i = 1 , as follows: γ b v b ρ b σ × ˚ v ( t , t , t ) = 12 t + 12 t ,γ b v b ρ b σ × ˚ v ( t , t , t ) = 12 t + 12 t + t . The resulting map g ǫ is illustrated in Figure 4 where it is exaggerated to show where each cellof Y ′ ǫ is sent.PSfrag replacements xyz . F IGURE