A Geometric Vietoris-Begle Theorem, with an Application to Riesz Spaces
aa r X i v : . [ m a t h . GN ] D ec A Geometric Vietoris-Begle Theorem, with an Application toRiesz Spaces
Andrew McLennanSchool of EconomicsThe University of Queensland ∗ December 17, 2019
Abstract
We show that a surjective map between compact absolute neighborhood retracts is a ho-motopy equivalence if the fibers are contractible and either the domain is simply connectedor the fibers are also absolute neighborhood retracts. This is a geometric analogue of theVietoris-Begle theorem. We use it to show that if L is a Hausdorff locally convex Riesz space, x ∈ L , the function y x ∨ y is continuous, and C ⊂ L is compact, convex, and metrizable,then { x ∨ y : y ∈ C } is a compact contractible absolute neighborhood retract. A metric space X is an absolute neighborhood retract (ANR) if, whenever Y is a metric space and A ⊂ Y is closed and homeomorphic to X , there is a neighborhood U ⊂ Y of A and a retraction r : U → A . These spaces were extensively studied from the time of their introduction by Borsuk(1932) until around 1970, but since that time they have become less well known. The property ofthese spaces used in our argument is developed at the beginning of Section 3. No other propertiesof these spaces are required before then, but possibly it would be good for the reader to be awarethat this class of spaces is quite general, encompassing both manifolds and simplicial complexes. ∗ Email: [email protected]. I am grateful to Rabee Tourky for guidance concerning Riesz spaces,and it is a pleasure to acknowledge a helpful conversation with Benjamin Burton.
1f course a continuous function f : X → Y is a homotopy equivalence if there is a continuous g : Y → X such that gf and f g are homotopic to the respective identities, and X and Y are homotopy equivalent if such f and g exist. The space X is contractible if there is a continuous c : X × [0 , → X such that c is the identity and c ( X ) is a singleton, which is to say that X ishomotopy equivalent to a point. Our first main results are: Theorem 1 If X and Y are compact connected ANR’s, f : X → Y is a continuous surjection,and, for each y ∈ Y , the fiber f − ( y ) is a contractible ANR, then f is a homotopy equivalence. Theorem 2 If X and Y are compact connected ANR’s, f : X → Y is a continuous surjection, X is simply connected, and, for each y ∈ Y , f − ( y ) is contractible, then f is a homotopy equivalence. These results are closely related to the Vietoris-Begle theorem. A space Z is acyclic withrespect to a homology (cohomology) theory if ˜ H n ( Z ) = 0 ( ˜ H n ( Z ) = 0) for all n = 0 , , , . . . .Since homology and cohomology are invariant under homotopy, a contractible space is acyclic forany homology or cohomology theory. Of the various versions of the Vietoris-Begle theorem inthe literature, we state two, which use Alexander-Spanier cohomology and homology respectively.The first might be regarded as the “standard” version. It asserts that if X and Y are paracompactHausdorff spaces, f : X → Y is a closed surjection, and, for some n ≥
0, ˜ H k ( f − ( y )) = 0 for all y ∈ Y and k < n , then ˜ H k ( f ) : ˜ H k ( X ) → ˜ H k ( Y ) is an isomorphism for k < n and an injection for k = n . A particularly elegant proof is given in Lawson (1973). The second is a dual result thatwas established by Volovikov and Anh (1984) and reproved by Dydak (1986). It asserts that if X and Y are compact metrizable spaces, f is a surjection, and, for some n ≥
0, ˜ H k ( f − ( y )) = 0 forall y ∈ Y and k < n , then ˜ H k ( f ) : ˜ H k ( X ) → ˜ H k ( Y ) is an isomorphism for k < n and a surjectionfor k = n . Evidently Theorems 1 and 2 and the Vietoris-Begle theorem say quite similar things,in geometric and algebraic language respectively.Theorems 1 and 2 have an interesting consequence. Let L be a Hausdorff locally convex Rieszspace. That is, L is a Hausdorff locally convex topological vector space over the reals endowedwith a partial order ≥ such that: a) for all x, y, z ∈ L and α ≥ x ≥ y implies that x + z ≥ y + z and αx ≥ αy ; b) any two elements x, y ∈ L have a least upper bound x ∨ y and a greatest lowerbound x ∧ y . For x ∈ L let u x , d x : L → L be the functions u x ( y ) = x ∨ y and d x ( y ) = x ∧ y . Theorem 3 If C ⊂ L is compact, convex, and metrizable, x ∈ L , and u x ( d x ) is continuous,then u x ( C ) ( d x ( C ) ) is a compact contractible ANR. It should be emphasized that this result is not easier to prove (so far as the author has beenable to determine) when L is finite dimensional.2or x ∈ L let | x | = ( x ∨ − ( x ∧ A ⊂ L is solid if, for all y ∈ A , A contains all x ∈ L such that | x | ≤ | y | , and L is locally solid if its topology has a base at the origin consistingof solid sets. A result of Roberts and Namioka (e.g., Aliprantis and Burkinshaw (2003), p. 55)asserts that a Riesz space is locally solid if and only if the function ( x, y ) x ∨ y is continuous,and this is the case if and only if ( x, y ) x ∧ y is continuous. An example on p. 56 of Aliprantisand Burkinshaw shows that this can fail to be the case even if u x and d x are continuous.Among various ways that C may be metrizable even if L is not, we mention that Varadarajan(1958) has shown that if L is the space of measures on a compact metric space with the weaktopology, then the lattice cone L + = { x ∈ L : x ≥ } is metrizable, but L is metrizable onlyunder quite restrictive conditions.In economics the existence of equilibrium is frequently proved by applying the Kakutani fixedpoint theorem and its infinite dimensional generalizations to upper hemicontinuous convex valuedcorrespondences. The author was led to wonder whether Theorem 3 might be true because itgives a method of passing from a convex valued correspondence to a contractible valued corre-spondence, to which the Eilenberg-Montgomery fixed point theorem (Eilenberg and Montgomery(1946)) might be applied. In this section X and Y are compact connected ANR’s and f : X → Y is a continuous surjection.The proofs of Theorems 1 and 2 verify, respectively, the hypotheses of the following two resultsof Whitehead (1948) (Theorem 1, p. 1133, and Theorem 3, p. 1135). (Actually the hypotheses ofWhitehead’s Theorem 1 are somewhat weaker.) Proposition 1 If π n ( f ) : π n ( X ) → π n ( Y ) is an isomorphism for all n , then f is a homotopyequivalence. Fix a point x ∈ X , and let y = f ( x ). Let ˜ X and ˜ Y be the universal covering spaces of X and Y , with respect to the base points x and y , and let ˜ f : ˜ X → ˜ Y be the lift of f with respectto these base points. Proposition 2 If π ( f ) : π ( X ) → π ( Y ) is an isomorphism and, for each n = 2 , , . . . , ˜ H n ( ˜ f ) :˜ H n ( ˜ X ) → ˜ H n ( ˜ Y ) is an isomorphism, then f is a homotopy equivalence. We present the proof of Theorem 2 first, because doing so allows some of the ideas andtechniques used in the proof of Theorem 1 to be introduced in a simplified setting. The followingfact will be used several times. 3 emma 1 If A is a compact topological space, B is a Hausdorff topological space, g : A → B iscontinuous, b ∈ B , and U ⊂ A is an open neighborhood of g − ( b ) , then there is an open V ⊂ B containing b such that g − ( V ) ⊂ U . Proof.
Otherwise for each open V ⊂ B containing b there would be an a V ∈ A \ U such that g ( a V ) ∈ V . The net { a V } would have subnet converging to a point in a ∈ A \ U because this setis compact. Since B is Hausdorff, continuity gives g ( a ) = b , but then a ∈ g − ( b ) ∩ ( A \ U ) = ∅ .Because they are metric spaces, X and Y can be embedded in Banach spaces L X and L Y (e.g.,McLennan (2018, Th. 6.3)). Since X and Y are ANR’s, there are retractions r X : U X → X and s Y : V Y → Y where U X ⊂ L X and V Y ⊂ L Y are neighborhoods of X and Y . Let W ⊂ Y × Y be aneighborhood of the diagonal { ( y, y ) : y ∈ Y } such that (1 − t ) y + ty ∈ V Y for all ( y , y ) ∈ W and t ∈ [0 , k let B k be the closed unit ball in R k , and let S k − be its boundary.A k -pair is a pair ( V ′ , V ) of open subsets of Y such that V ′ ⊂ V , V × V ⊂ W , and for any η ∂ : S k − → f − ( V ′ ) there is a continuous η : B k → f − ( V ) such that η | S k − = η ∂ . A k -collection is a collection V k of k -pairs such that { V ′ : ( V ′ , V ) ∈ V k } is a cover of Y . Lemma 2
If each f − ( y ) is path connected, then, for any open V ⊂ Y such that V × V ⊂ W and any y ∈ V , there is an open V ′ ⊂ V containing y such that ( V ′ , V ) is a -pair. Proof.
Since f − ( y ) is compact there is a δ > δ -ball B δ in L X around f − ( y ) is contained in r − X ( f − ( V )). Lemma 1 gives an open V ′ ⊂ V containing y such that f − ( V ′ ) ⊂ B δ . Suppose η ∂ is a function from {− , } to f − ( V ′ ). Choose x − , x ∈ f − ( y )such that the distance from η ∂ ( −
1) to x − and the distance from η ∂ (1) to x are both less than δ . Let π : [0 , → f − ( y ) be a continuous path with π ( −
1) = x − and π (1) = x . Define η : [ − , → f − ( V ′ ) by setting η ( t ) = r X (( − t − η ∂ ( −
1) + (2 t + 2) x − ) , − ≤ t ≤ − ,π ( t + ) , − ≤ t ≤ ,r X ((2 − t ) x + (2 t − η ∂ (1)) , ≤ t ≤ . Proposition 3
If, for each y ∈ Y , f − ( y ) is path connected, then π ( f ) is surjective. Proof.
Lemma 2 implies that there is a 1-collection V . The Lebesgue number lemma gives an ε > y ∈ Y there is some ( V ′ , V ) ∈ V such that V ′ contains the ε -ballcentered at y . 4ix a map g : S → Y . We will show that g approximately lifts to X in the sense that thereis a γ : S → X such that ( g ( p ) , f ( γ ( p ))) ∈ W for all p ∈ S . Then h ( p, t ) = s Y ((1 − t ) g ( t ) + tf ( γ ( p ))) is a homotopy between g and f γ , so the homotopy class of g is in the image of π ( f ).It is easy to produce a triangulation T of S such that for any 1-simplex σ ∈ T , the diameter of g ( σ ) is less than ε . For each k = 0 , T k be the set of k -simplices in T , and let T ( k ) = S σ ∈ T k σ be the k -skeleton of T . Construct a map γ : T (0) → X by letting each γ ( v ) be an element of f − ( g ( v )). For each σ ∈ T there is some ( V ′ , V ) ∈ V k such that g ( σ ) ⊂ V ′ and γ ( ∂σ ) ⊂ f − ( V ).Consequently we can define γ | σ to be an extension of γ | ∂σ such that γ ( σ ) ⊂ f − ( V ), whichgives ( g ( p ) , f ( γ ( p ))) ⊂ V × V ⊂ W for all p ∈ σ .It is now easy to prove Theorem 2. The last result implies that π ( f ) is surjective, and since X is simply connected, it follows that Y is simply connected, so ˜ X , ˜ Y , and ˜ f are (up to irrelevantformalities) just X , Y , and f . As we mentioned previously, since each fibre f − ( y ) is contractible,it is acyclic, and ˜ X = X is compact, so the dual Vietoris-Begle theorem of Volovikov-Ahn andDydak implies that ˜ H n ( ˜ f ) is an isomorphism for all n ≥
2, after which Proposition 2 impliesthat f is a homotopy equivalence. (The dual Vietoris-Begle theorem is specific to Alexander-Spanier homology, and Whitehead does not specify which homology theory he is using. However,it is well known that Alexander-Spanier homology agrees with ˇCech homology on compact Haus-dorff spaces, and ˇCech and singular homology agree on ANR’s (Dugundji, 1955, Kodama, 1955,Mardeˇsi´c, 1958).)In the proof of Theorem 1 the next two results play the roles played by Lemma 2 and Propo-sition 3 in the proof of Theorem 2. Lemma 3 If V ⊂ Y is open, V × V ⊂ W , y ∈ V , and f − ( y ) is a contractible ANR, then thereis an open V ′ ⊂ V containing y such that ( V ′ , V ) is a k -pair for any positive integer k . Proof.
Since f − ( y ) is an ANR there is a neighborhood U y ⊂ X of f − ( y ) and a retraction r y : U y → f − ( y ). Since f − ( y ) is compact, U ′ y = { x ∈ U y : (1 − t ) x + tr y ( x ) ∈ r − X ( f − ( V )) for all t ∈ [0 , } is an open neighborhood of f − ( y ). Lemma 1 implies that there is an open V ′ ⊂ V containing y such that f − ( V ′ ) ⊂ U ′ y . Let c y : f − ( y ) × [0 , → f − ( y ) be a contraction. Using therepresentation of points in B k as products tp of scalars t ∈ [0 ,
1] and points p ∈ S k − , for a givencontinuous η ∂ : S k − → f − ( V ′ ) we define an extension η : B k → f − ( V ) by setting η ( tp ) = r X (cid:0) (2 t − η ∂ ( p ) + 2(1 − t ) r y ( η ∂ ( p )) (cid:1) , ≥ t ≥ ,c y (cid:0) r y ( η ∂ ( p )) , − t (cid:1) , ≥ t ≥ . Proposition 4
If, for each y ∈ Y , f − ( y ) is a contractible ANR, and n ≥ , then π n − ( f ) isinjective and π n ( f ) is surjective. Proof.
Let M = S n − × [0 ,
1] and ∂M = S n − × { , } , and let η : ∂M → X and g : M → Y begiven maps such that g | ∂M = f η . We will show that g approximately lifts to X in the sense thatthere is a continuous extension γ n : M → X of η such that ( g ( p ) , f ( γ n ( p ))) ∈ W for all p ∈ M , sothat h ( p, t ) = s Y ((1 − t ) g ( t ) + tf ( γ n ( p ))) is a homotopy between g and f γ n . This evidently impliesthat π n − ( f ) is injective. It will be easy to see that a similar, slightly simpler, argument showsthat any given g : S n → Y approximately lifts to X in the sense that there is a γ n : S n → X such that ( g ( p ) , f ( γ n ( p ))) ∈ W for all p ∈ S n , so that h ( p, t ) = s Y ((1 − t ) g ( t ) + tf ( γ n ( p ))) is ahomotopy between g and f γ n , and of course this implies that π n ( f ) is surjective.We begin by constructing k -collections V k for k = 1 , . . . , n and numbers ε , . . . , ε n > k and y ∈ Y there is some ( V ′ , V ) ∈ V k such that V ′ contains the 2 ε k -ball centeredat y .(b) For each k = 2 , . . . n and ( V ′ , V ) ∈ V k − the diameter of V is less than ε k / n -collection V n . Proceeding by descending induction, supposethat for some k = 2 , . . . , n we have already defined V k . The Lebesgue number lemma gives an ε k > y ∈ Y there is some ( V ′ , V ) ∈ V k such that V ′ contains the 2 ε k -ballcentered at y . Lemma 3 implies the existence of a ( k − V k − such that for each( V ′ , V ) ∈ V k − the diameter of V is less than ε k /
2. After this inductive construction the finalstep is to observe that the Lebesgue number lemma gives an ε > y ∈ Y there is some ( V ′ , V ) ∈ V such that V ′ contains the ε -ball centered at y .It is easy to construct a triangulation T of M . For such a T and k = 0 , . . . , n let T k be theset of k -simplices of T , and let T ( k ) = S ℓ ≤ k,σ ∈ T ℓ σ be the k -skeleton of T . Repeated barycentricsubdivision (e.g., Dold (1980)) eventually gives a triangulation T of M such that for each k =1 , . . . , n and σ ∈ T k , the diameter of g ( σ ) is less than ε k and consequently there is a ( V ′ , V ) ∈ V k such that the open ball of radius ε k around g ( σ ) is contained in V ′ . We will construct a continuousextension γ n : M → X of η such that for each σ ∈ T n there is a ( V ′ , V ) ∈ V n such that g ( σ ) ⊂ V ′ and γ n ( σ ) ⊂ f − ( V ). Since V ′ × V ⊂ W it follows that ( g ( p ) , f ( γ n ( p ))) ∈ W for all p ∈ M .6o construct an extension γ : T (0) ∪ ∂M → X of η , for each v ∈ T with v / ∈ ∂M let γ ( v ) bean element of f − ( g ( v )). Proceeding by ascending induction, suppose that for some k = 1 , . . . , n we have already defined a continuous extension γ k − : T ( k − ∪ ∂M → X such that for each τ ∈ T k − there is some ( V ′ , V ) ∈ V k − such that g ( τ ) ⊂ V ′ and γ k − ( τ ) ⊂ f − ( V ). Fix σ ∈ T k ,and choose ( V ′ , V ) ∈ V k such that the open ball of radius ε k around g ( σ ) is contained in V ′ . If σ ⊂ ∂M , then η ( σ ) ⊂ f − ( g ( σ )) ⊂ f − ( V ′ ).Otherwise for each facet τ of σ there is a pair ( V ′ τ , V τ ) ∈ V k − such that g ( τ ) ⊂ V ′ τ and γ k − ( τ ) ⊂ f − ( V τ ), and by (b) the diameter of V τ is less than ε k /
2. It follows that g ( ∂σ ) iscontained in the ε k -ball centered at g ( v ), so g ( ∂σ ) ⊂ V ′ and γ k − ( ∂σ ) ⊂ f − ( V ′ ). Consequentlywe can let γ k | σ be an extension of γ k − | ∂σ such that γ k ( σ ) ⊂ f − ( V ). Doing this for all σ ∈ T k gives a satisfactory extension γ k : T ( k ) ∪ ∂M → X . We begin with a brief review of the key sufficient condition for a space to be an ANR. A gen-eralization of the Tietze extension theorem due to Dugundji (1951) asserts that if X is a metricspace, A ⊂ X is closed, Y is a Hausdorff locally convex topological vector space, and f : A → Y is continuous, then there is a continuous extension f : X → Y whose image is contained in theconvex hull of f ( A ). A proof can be found in McLennan (2018, pp. 138–9), but it is a variantof the final argument of this section, and the reader may have little difficulty seeing how, withsuitable modifications, this can do the job. Lemma 4 If Y is a Hausdorff locally convex topological vector space, C ⊂ Y is convex, U ⊂ C is (relatively) open, A ⊂ U is metrizable, and r : U → A is a retraction, then A is an ANR. Proof.
Suppose that X is a metric space and h : A → X maps A homeomorphically ontoits image, which is closed. Dugundji’s theorem implies that h − : h ( A ) → A has a continuousextension j : X → C . Let V = j − ( U ). Of course V is a neighborhood of h ( A ), and h ◦ r ◦ j | V : V → h ( A ) is a retraction.We now turn to the setting of Theorem 3: L is a Hausdorff locally convex Riesz space, x ∈ L , u x is continuous, and C ⊂ L is compact, convex, and metrizable. Let D = u x ( C ). The remainderof this section proves that D is a compact contractible ANR. (Exactly the same argument showsthat d x ( C ) is a compact contractible ANR when d x is continuous.) Of course D is compact andconnected because it is the continuous image of a connected compact space.It suffices to prove the claim with x = 0, because C − x is homeomorphic to C , u x ( C ) =7 ( C − x ) + x , and a translate of a contractible ANR is a contractible ANR. Thus we assume that x = 0, and we write u in place of u .The Riesz decomposition property asserts that if x , x , y ≥ y ≤ x + x , then there are y ≥ y ≥ y ≤ x , y ≤ x , and y + y = y . (To prove this let y = y ∧ x and y = y − y . Clearly y , y ≥ y + y = y . Finally y = y − y ∧ x = 0 ∨ ( y − x ) ≤ ∨ x = x .)We claim that the fiber u − ( y ) above each y ∈ D is convex. Suppose that x , x ∈ C , u ( x ) = y = u ( x ), and 0 < α <
1. Of course y ≥ y = (1 − α ) y + αy ≥ (1 − α ) x + αx , so y is an upperbound of { , (1 − α ) x + αx } , and we need to show that it is the least upper bound. Suppose that y ≥ y ′ ≥ y ′ ≥ (1 − α ) x + αx . Then 0 ≤ y − y ′ ≤ (1 − α )( y − x ) + α ( y − x ), and the Rieszdecomposition property gives w , w such that 0 ≤ w ≤ (1 − α )( y − x ), 0 ≤ w ≤ α ( y − x ),and w + w = y − y ′ . We have x ≤ y − w / (1 − α ) ≤ y − w , and y − w ≥ w ≤ w + w = y − y ′ ≤ y . Thus y − w is an upper bound of { , x } , which contradicts x ∨ y if w = 0, so w = 0. Symmetrically, w = 0, so y ′ = y .Since L is locally convex and C is metrizable, Lemma 4 implies that C and the fibers u − ( y ) areANR’s. Of course the fibers are contractible because they are convex, and C is simply connectedbecause it is convex. The remaining hypothesis of Theorems 1 and 2 is that D is an ANR, soonce we have verified this, either of these results will imply that u is a homotopy equivalence, andthus that D is homotopy equivalent to a convex set.The Urysohn metrization theorem (Kelley, 1955, p. 125) asserts that a regular T space ismetrizable if it has a countable base. By assumption L is Hausdorff, hence T . As a topologicalvector space, L is regular (Schaefer, 1999, p. 16). These properties are inherited by subspaces,so C and D are regular and T . Since C is compact and metrizable, it is easy to construct acountable base for it. Consider y ∈ D and a neighborhood V of y . Since f − ( y ) is compact, it iscovered by a finite union of base sets for C that are contained in f − ( V ). Lemma 1 gives an open V ′ ⊂ V containing y such that f − ( V ′ ) is contained in this finite union, and thus the interior ofthe image of this finite union is contained in V and contains y . We have shown that the set ofinteriors of images of finite unions of base sets for C is a base for D , so D has a countable baseand is thus metrizable.Standard results imply that, since D is metrizable, it can be embedded as a closed subset ofa convex subset E of a Banach space (McLennan, 2018, Th. 6.3). In view of Lemma 4 it sufficesto show that D is a retract of E . For each w ∈ E \ D let B w be the open ball in E centered at w whose radius is one half of the distance from w to D . Since metric spaces are paracompact,the open cover { B w ∩ E } of E \ D has a locally finite refinement U . For each U ∈ U choose an x U ∈ C such that the distance from U to u ( x U ) is less than twice the distance from U to D . Let8 ϕ U } U ∈U be a partition of unity subordinate to U . Define ρ : E \ D → D by setting ρ ( z ) = u (cid:0) X U ϕ U ( z ) x U (cid:1) . Let r : E → D be the function that is the identity on D and ρ on E \ D . Evidently r is a retractionif we can show that it is continuous. Since D is closed in E , r is continuous at each point in E \ D .Consider a point y ∈ D and a neighborhood V ⊂ D . We need to find a neighborhood V ′′ ⊂ E of y such that r ( V ′′ ) ⊂ V .Let W be a convex neighborhood of u − ( y ) that is contained in u − ( V ). (To prove that sucha W exists consider that, because L is locally convex, for each x ∈ u − ( y ) there is a convexneighborhood A x of the origin such that ( x + 2 A x ) ∩ C ⊂ u − ( V ). If x + A x , . . . , x k + A x k isa finite subcover of { x + A x : x ∈ u − ( y ) } and A = T i A x i , then ( u − ( y ) + A ) ∩ C ⊂ u − ( V ).)Lemma 1 gives a neighborhood V ′ ⊂ V of y such that u − ( V ′ ) ⊂ W . Let δ > V ′ contains the ball of radius δ (in D ) centered at y , and let V ′′ be the ball of radius δ/ E ) centered at y . Of course r ( D ∩ V ′′ ) = D ∩ V ′′ ⊂ V ′ ⊂ V .Consider a point z ∈ V ′′ \ D . If z ∈ U ∈ U , then U is contained in some B w whose radius is lessthan the distance from D to B w , which is less than the distance from y to z , so B w is containedin the ball of radius 3 δ/ y . The distance from U to u ( x U ) is also less than twice thedistance from y to z , so it is less than 2 δ/
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