Parametrized Borsuk-Ulam problem for projective space bundles
aa r X i v : . [ m a t h . A T ] S e p PARAMETRIZED BORSUK-ULAM PROBLEM FORPROJECTIVE SPACE BUNDLES
MAHENDER SINGH
Abstract.
Let π : E → B be a fiber bundle with fiber having the mod2 cohomology algebra of a real or a complex projective space and let π ′ : E ′ → B be vector bundle such that Z acts fiber preserving andfreely on E and E ′ −
0, where 0 stands for the zero section of the bundle π ′ : E ′ → B . For a fiber preserving Z -equivariant map f : E → E ′ ,we estimate the cohomological dimension of the zero set Z f = { x ∈ E | f ( x ) = 0 } . As an application, we also estimate the cohomologicaldimension of the Z -coincidence set A f = { x ∈ E | f ( x ) = f ( T ( x )) } ofa fiber preserving map f : E → E ′ . Introduction
The unit n -sphere S n is equipped with the antipodal involution given by x
7→ − x . The well known Borsuk-Ulam theorem states that: If n ≥ k , thenfor every continuous map f : S n → R k there exists a point x in S n suchthat f ( x ) = f ( − x ). Over the years there have been several generalizationsof the theorem in many directions. We refer the reader to the article [15] bySteinlein which lists 457 publications concerned with various generalizationsof the Borsuk-Ulam theorem.Jaworowski [5], Dold [2], Nakaoka [12] and others extended this theoremto the setting of fiber bundles, by considering fiber preserving maps f : SE → E ′ , where SE denotes the total space of the sphere bundle SE → B associated to a vector bundle E → B , and E ′ → B is other vectorbundle. Thus, they parametrized the Borsuk-Ulam theorem, whose generalformulation is as follows: Let G be a compact Lie group. Consider a fiber bundle π : E → B and avector bundle π ′ : E ′ → B such that G acts fiber preserving and freely on E and E ′ − , where stands for the zero section of the bundle π ′ : E ′ → B .For a fiber preserving G -equivariant map f : E → E ′ , the parametrizedversion of the Borsuk-Ulam theorem deals in estimating the cohomological Mathematics Subject Classification.
Primary 55M20; Secondary 55R91, 55R25.
Key words and phrases.
Characteristic polynomial of a bundle, cohomological dimen-sion, continuity of ˇCech cohomology, equivariant map, free involution. dimension of the zero set Z f = { x ∈ E | f ( x ) = 0 } . Such results appeared first in the papers of Jaworowski [5], Dold [2] andNakaoka [12]. Dold [2] and Nakaoka [12] defined certain polynomials, whichthey called the characteristic polynomials, for vector bundles with free G -actions ( G = Z p or S ) and used them for obtaining such results. Thecharacteristic polynomials were used by Koikara and Mukerjee [9] to provea parametrized version of the Borsuk-Ulam theorem for bundles whose fiberis a product of spheres, with the free involution given by the product of theantipodal involutions. Recently, Mattos and Santos [10] also used the sametechnique to obtain parametrized Borsuk-Ulam theorems for bundles whosefiber has the mod p cohomology algebra (with p >
2) of a product of twospheres with any free Z p -action and for bundles whose fiber has the rationalcohomology algebra of a product of two spheres with any free S -action.Jaworowski obtained parametrized Borsuk-Ulam theorems for lens spacebundles in [8] and parametrized Borsuk-Ulam theorems for sphere bundlesin [5, 6, 7].The purpose of this paper is to prove parametrized Borsuk-Ulam theo-rems for bundles whose fiber has the mod 2 cohomology algebra of a realor a complex projective space with any free involution. The theorems arestated in section 4 and proved in section 6. As an application, in section 7,the cohomological dimension of the Z -coincidence set of a fiber preservingmap is also estimated. 2. Preliminaries
Here we recall some basic notions that will be used in later sections.All spaces under consideration will be paracompact Hausdorff spaces andthe cohomology used will be the ˇCech cohomology with Z coefficients. Wewill exploit the continuity property of the ˇCech cohomology theory, for thedetails of which we refer to Eilenberg-Steenrod [3, Chapter X].We recall that a finitistic space is a paracompact Hausdorff space whoseevery open covering has a finite dimensional open refinement, where thedimension of a covering is one less than the maximum number of membersof the covering which intersect non-trivially (the notion was introduced bySwan in [16]). It is a large class of spaces including all compact Hausdorffspaces and all paracompact spaces of finite covering dimension.For a space X , cohom.dim ( X ) will mean the cohomological dimensionof X with respect to Z . For basic results of dimension theory, we refer to ARAMETRIZED BORSUK-ULAM PROBLEM 3
Nagami [11]. If G is a compact Lie group acting freely on a paracompactHausdorff space X , then X → X/G is a principal G -bundle and we can take a classifying map X/G → B G for the principal G -bundle X → X/G , where B G is the classifying space ofthe group G . Recall that for G = Z , we have H ∗ ( B G ; Z ) ∼ = Z [ s ] , where s is a homogeneous element of degree one. We will also use someelementary notions about vector bundles for the details of which we referto Husemoller [4].3. Free involutions on projective spaces and their orbitspaces
We note that odd dimensional real projective spaces admit free involu-tions. Let n = 2 m − m ≥
1. Recall that R P n is the orbit space ofthe antipodal involution on S n given by( x , x , ..., x m − , x m ) ( − x , − x , ..., − x m − , − x m ) . If we denote an element of R P n by [ x , x , ..., x m − , x m ], then the map R P n → R P n given by[ x , x , ..., x m − , x m ] [ − x , x , ..., − x m , x m − ]defines an involution. If[ x , x , ..., x m − , x m ] = [ − x , x , ..., − x m , x m − ] , then ( − x , − x , ..., − x m − , − x m ) = ( − x , x , ..., − x m , x m − ) , which gives x = x = ... = x m − = x m = 0, a contradiction. Hence, theinvolution is free.Similarly, the complex projective space C P n admit free involutions when n ≥ C P n is the orbit space of the free S -action on S n +1 given by( z , z , ..., z n , z n +1 ) ( ζ z , ζ z , ..., ζ z n , ζ z n +1 ) for ζ ∈ S . If we denote an element of C P n by [ z , z , ..., z n , z n +1 ], then the map[ z , z , ..., z n , z n +1 ] [ − z , z , ..., − z n +1 , z n ] MAHENDER SINGH defines an involution on C P n . If[ z , z , ..., z n , z n +1 ] = [ − z , z , ..., − z n +1 , z n ] , then ( λz , λz , ..., λz n , λz n +1 ) = ( − z , z , ..., − z n +1 , z n )for some λ ∈ S , which gives z = z = ... = z n = z n +1 = 0, a contradiction.Hence, the involution is free.We write X ≃ R P n if X is a space having the mod 2 cohomologyalgebra of R P n . Similarly, we write X ≃ C P n if X is a space having themod 2 cohomology algebra of C P n .Recently, Singh [14] determined completely the mod 2 cohomology al-gebra of orbit spaces of free involutions on mod 2 cohomology real andcomplex projective spaces. Using the Leray spectral sequence associated tothe Borel fibration X ֒ → X G −→ B G , they proved the following results. Theorem 3.1. If G = Z acts freely on a finitistic space X ≃ R P n , where n is odd, then H ∗ ( X/G ; Z ) ∼ = Z [ u, v ] / h u , v n +12 i , where deg( u )= and deg( v )= . Theorem 3.2. If G = Z acts freely on a finitistic space X ≃ C P n , where n is odd, then H ∗ ( X/G ; Z ) ∼ = Z [ u, v ] / h u , v n +12 i , where deg( u )= and deg( v )= .Remark . It is easy to see that, when n is even, then Z cannot act freelyon a finitistic space X ≃ R P n or C P n . For, if n is even, then the Eulercharacteristic χ ( X ) = (cid:26) X ≃ R P n n + 1 when X ≃ C P n .But for a free involution, χ ( X Z ) = 0 and hence the Floyd’s Euler charac-teristic formula [1, p.145] χ ( X ) + χ ( X Z ) = 2 χ ( X/ Z )gives a contradiction. Remark . Let X ≃ H P n be a finitistic space, where H P n is the quater-nionic projective space. For n = 1, X ≃ S , which is dealt in [2]. For n ≥ X , which follows from the stronger fact thatsuch spaces have the fixed point property. ARAMETRIZED BORSUK-ULAM PROBLEM 5
Remark . Let X ≃ O P be a finitistic space, where O P is the Cayleyprojective plane. Note that H ∗ ( O P ; Z ) ∼ = Z [ u ] / h u i , where u is a ho-mogeneous element of degree 8. Just as in Remark 3.3, it follows from theFloyd’s Euler characteristic formula that there is no free involution on X .4. Statements of theorems
Let X ≃ R P n be a finitistic space. Let ( X, E, π, B ) be a fiber bun-dle with a fiber preserving free Z -action such that the quotient bundle( X/G, E, π, B ) has a cohomology extension of the fiber, that is, there is a Z -module homomorphism of degree zero θ : H ∗ ( X/G ; Z ) → H ∗ ( E ; Z )such that for any b ∈ B , the composition H ∗ ( X/G ; Z ) θ → H ∗ ( E ; Z ) i ∗ b → H ∗ (( X/G ) b ; Z )is an isomorphism, where i b : ( X/G ) b ֒ → E is the inclusion of the fiber over b (see [1, p.372]). This condition on thebundle is assumed so that the Leray-Hirsch theorem can be applied (see [1,Chapter VII, Theorem 1.4]). Now consider a k -dimensional vector bundle π ′ : E ′ → B with a fiber preserving Z -action on E ′ which is free on E ′ −
0. Let f : E → E ′ be a fiber preserving Z -equivariant map. Define Z f = { x ∈ E | f ( x ) = 0 } and Z f = Z f / Z , the quotient by the free Z -action induced on Z f .Let H ∗ ( B )[ x, y ] be the polynomial ring over H ∗ ( B ) in the indeterminates x and y . For the bundle ( X ≃ R P n , E, π, B ), in section 5, we will define thecharacteristic polynomials W ( x, y ) and W ( x, y ) in H ∗ ( B )[ x, y ] and we willshow that H ∗ ( E ) and H ∗ ( B )[ x, y ] / h W ( x, y ) , W ( x, y ) i are isomorphic as H ∗ ( B )-modules. Therefore, each polynomial q ( x, y ) in H ∗ ( B )[ x, y ] definesan element of H ∗ ( E ), which we will denote by q ( x, y ) | E . We will denote by q ( x, y ) | Z f MAHENDER SINGH the image of q ( x, y ) | E by the H ∗ ( B )-homomorphism i ∗ : H ∗ ( E ) → H ∗ ( Z f ) , where i ∗ is the map induced by the inclusion i : Z f ֒ → E . In a similar way,we will define the characteristic polynomial W ′ ( x ) for the vector bundle π ′ : E ′ → B . With the above hypothesis and notations, we prove the followingresults for the real case. Theorem 4.1.
Let X ≃ R P n be a finitistic space. If q ( x, y ) in H ∗ ( B )[ x, y ] is a polynomial such that q ( x, y ) | Z f = 0 , then there are polynomials r ( x, y ) and r ( x, y ) in H ∗ ( B )[ x, y ] such that q ( x, y ) W ′ ( x ) = r ( x, y ) W ( x, y ) + r ( x, y ) W ( x, y ) in the ring H ∗ ( B )[ x, y ] , where W ′ ( x ) , W ( x, y ) and W ( x, y ) are the char-acteristic polynomials. As a corollary, just as in [2], we have the following parametrized versionof the Borsuk-Ulam theorem.
Corollary 4.2.
Let X ≃ R P n be a finitistic space. If the fiber dimensionof E ′ → B is k , then q ( x, y ) | Z f = 0 for all non zero polynomials q ( x, y ) in H ∗ ( B )[ x, y ] , whose degree in x and y is less than ( n − k + 1) . Equivalently,the H ∗ ( B ) -homomorphism n − k M i + j =0 H ∗ ( B ) x i y j → H ∗ ( Z f ) given by x i → x i | Z f and y j → y j | Z f is a monomorphism. As a result, if n ≥ k , then cohom.dim ( Z f ) ≥ cohom.dim ( B ) + ( n − k ) . Let X ≃ C P n be a finitistic space. Just as in the real case, we will definethe characteristic polynomials W ( x, y ) and W ( x ) for the bundle ( X ≃ C P n , E, π, B ) and show that H ∗ ( E ) and H ∗ ( B )[ x, y ] / h W ( x, y ) , W ( x ) i areisomorphic as H ∗ ( B )-modules. With similar hypothesis and notations as inthe real case, we prove the following results for the complex case. Theorem 4.3.
Let X ≃ C P n be a finitistic space. If q ( x, y ) in H ∗ ( B )[ x, y ] is a polynomial such that q ( x, y ) | Z f = 0 , then there are polynomials r ( x, y ) and r ( x, y ) in H ∗ ( B )[ x, y ] such that q ( x, y ) W ′ ( x ) = r ( x, y ) W ( x, y ) + r ( x, y ) W ( x ) ARAMETRIZED BORSUK-ULAM PROBLEM 7 in the ring H ∗ ( B )[ x, y ] , where W ′ ( x ) , W ( x, y ) and W ( x ) are the charac-teristic polynomials. Corollary 4.4.
Let X ≃ C P n be a finitistic space. If the fiber dimensionof E ′ → B is k , then q ( x, y ) | Z f = 0 for all non zero polynomials q ( x, y ) in H ∗ ( B )[ x, y ] , whose degree in x and y is less than (2 n − k + 2) . Equivalently,the H ∗ ( B ) -homomorphism n − k +1 M i + j =0 H ∗ ( B ) x i y j → H ∗ ( Z f ) given by x i → x i | Z f and y j → y j | Z f is a monomorphism. As a result, if n ≥ k , then cohom.dim ( Z f ) ≥ cohom.dim ( B ) + (2 n − k + 1) . Characteristic polynomials for bundles
Let (
X, E, π, B ) be a fiber bundle with a fiber preserving free Z -actionsuch that the quotient bundle ( X/G, E, π, B ) has a cohomology extensionof the fiber. With this hypothesis, we now proceed to define the character-istic polynomials for the bundles. We deal the real and the complex caseseparately.
When X ≃ R P n . Let G = Z act freely on a finitistic space X ≃ R P n . Then, n is oddand by theorem 3.1, H ∗ ( X/G ; Z ) is a free graded algebra generated by theelements 1 , u, v, uv, ..., v n − , uv n − , subject to the relations u = 0 and v n +12 = 0, where u ∈ H ( X/G ; Z ) and v ∈ H ( X/G ; Z ). Let ( X ≃ R P n , E, π, B ) be a bundle with the hypothesisof section 4. By the Leray-Hirsch theorem, there exist elements a ∈ H ( E )and b ∈ H ( E ) such that the restriction to a typical fiber j ∗ : H ∗ ( E ) → H ∗ ( X/G )maps a u and b v . Note that H ∗ ( E ) is a H ∗ ( B )-module, via theinduced homomorphism π ∗ and is generated by the basis(5.1) 1 , a, b, ab, ..., b n − , ab n − . We can express the element b n +12 ∈ H n +1 ( E ) in terms of the basis (5.1).Therefore, there exist unique elements w i ∈ H i ( B ) such that b n +12 = w n +1 + w n a + w n − b + · · · + w b n − + w ab n − . MAHENDER SINGH
Similarly, we express the element a ∈ H ( E ) as a = ν + ν a + αb, where ν i ∈ H i ( B ) and α ∈ Z are unique elements. The characteristicpolynomials in the indeterminates x and y , of degrees respectively 1 and 2,associated to the fiber bundle ( X ≃ R P n , E, π, B ) are defined by W ( x, y ) = w n +1 + w n x + w n − y + · · · + w y n − + w xy n − + y n +12 and W ( x, y ) = ν + ν x + αy + x . On substituting the values for the indeterminates x and y , we obtain thefollowing homomorphism of H ∗ ( B )-algebras σ : H ∗ ( B )[ x, y ] → H ∗ ( E )given by ( x, y ) ( a, b ). The Ker ( σ ) is the ideal generated by the polyno-mials W ( x, y ) and W ( x, y ) and hence(5.2) H ∗ ( B )[ x, y ] / h W ( x, y ) , W ( x, y ) i ∼ = H ∗ ( E ) . When X ≃ C P n . Since this case is similar, we present it rather briefly. Let G = Z actfreely on a finitistic space X ≃ C P n . Then, n is odd and by theorem 3.2, H ∗ ( X/G ; Z ) is a free graded algebra generated by the elements1 , u, u , v, uv, ..., v n − , uv n − , u v n − , subject to the relations u = 0 and v n +12 = 0, where u ∈ H ( X/G ; Z )and v ∈ H ( X/G ; Z ). By the Leray-Hirsch theorem, there exist elements a ∈ H ( E ) and b ∈ H ( E ) such that the restriction to a typical fiber j ∗ : H ∗ ( E ) → H ∗ ( X/G )maps a u and b v . Note that H ∗ ( E ) is a H ∗ ( B )-module and isgenerated by the basis(5.3) 1 , a, a , b, ab, ..., b n − , ab n − , a b n − . We write b n +12 ∈ H n +2 ( E ) in terms of the basis (5.3). Thus, there existunique elements w i ∈ H i ( B ) such that b n +12 = w n +2 + w n +1 a + w n a + · · · + w a b n − . Similarly, we write the element a ∈ H ( E ) as a = ν + ν a + ν a , ARAMETRIZED BORSUK-ULAM PROBLEM 9 where ν i ∈ H i ( B ) are unique elements. The characteristic polynomials inthe indeterminates x and y , of degrees respectively 1 and 4, associated tothe fiber bundle ( X ≃ C P n , E, π, B ) are defined by W ( x, y ) = w n +2 + w n +1 x + w n x + · · · + w x y n − + y n +12 and W ( x ) = ν + ν x + ν x + x . This gives a homomorphism of H ∗ ( B )-algebras σ : H ∗ ( B )[ x, y ] → H ∗ ( E )given by ( x, y ) ( a, b ) and having Ker ( σ ) as the ideal generated by thepolynomials W ( x, y ) and W ( x ). Hence(5.4) H ∗ ( B )[ x, y ] / h W ( x, y ) , W ( x ) i ∼ = H ∗ ( E ) . Characteristic polynomial for the bundle π ′ : E ′ → B . Now we define the characteristic polynomial associated to the k -dimensionalvector bundle π ′ : E ′ → B with fiber preserving Z -action on E ′ which is freeon E ′ −
0. Let SE ′ denote the total space of sphere bundle of π ′ : E ′ → B . Since the action is free on SE ′ , we obtain the projective space bundle( R P k − , SE ′ , π ′ , B ) and the principal Z -bundle SE ′ → SE ′ . We know that H ∗ ( R P k − ; Z ) ∼ = Z [ u ′ ] / h u ′ k i , where u ′ = g ∗ ( s ), s ∈ H ( B G ) and g : R P k − → B G is a classifying map forthe principal Z -bundle S k − → R P k − . Let h : SE ′ → B G be a classifyingmap for the principal Z -bundle SE ′ → SE ′ and let a ′ = h ∗ ( s ) ∈ H ( SE ′ ).Now the Z -module homomorphism θ ′ : H ∗ ( R P k − ) → H ∗ ( SE ′ )given by u ′ a ′ is a cohomology extension of the fiber. Again, by the Leray-Hirsch theorem H ∗ ( SE ′ ) is a H ∗ ( B )-module via the induced homomorphism π ′ ∗ and is generated by the basis1 , a ′ , a ′ , ..., a ′ k − . We write a ′ k ∈ H k ( SE ′ ) as a ′ k = w ′ k + w ′ k − a ′ + · · · + w ′ a ′ k − , where w ′ i ∈ H i ( B ) are unique elements. Now the characteristic polynomialin the indeterminate x of degree 1, associated to the vector bundle π ′ : E ′ → B is defined as W ′ ( x ) = w ′ k + w ′ k − x + · · · + w ′ x k − + x k . By similar arguements as used above, we have the following isomorphism of H ∗ ( B )-algebras H ∗ ( B )[ x ] / h W ′ ( x ) i ∼ = H ∗ ( SE ′ )given by x a ′ . Proofs of theorems
We first prove our results for the real case.
Proof of Theorem 4.1.
Let q ( x, y ) in H ∗ ( B )[ x, y ] be a polynomial such that q ( x, y ) | Z f = 0. It follows from the continuity property of the ˇCech coho-mology theory, that there is an open subset V ⊂ E such that Z f ⊂ V and q ( x, y ) | V = 0. Consider the long exact cohomology sequence for the pair( E, V ), namely, · · · → H ∗ ( E, V ) j ∗ → H ∗ ( E ) → H ∗ ( V ) → H ∗ ( E, V ) → · · · . By exactness, there exists µ ∈ H ∗ ( E, V ) such that j ∗ ( µ ) = q ( x, y ) | E , where j : E → ( E, V )is the natural inclusion. The Z -equivariant map f : E → E ′ gives the map f : E − Z f → E ′ − . The induced map f ∗ : H ∗ ( E ′ − → H ∗ ( E − Z f )is a H ∗ ( B )-homomorphism. Also we have W ′ ( a ′ ) = 0. Therefore, W ′ ( x ) | E − Z f = W ′ ( a ) = W ′ (cid:0) f ∗ ( a ′ ) (cid:1) = f ∗ (cid:0) W ′ ( a ′ ) (cid:1) = 0 . Now consider the long exact cohomology sequence for the pair (
E, E − Z f ),that is, · · · → H ∗ ( E, E − Z f ) j ∗ → H ∗ ( E ) → H ∗ ( E − Z f ) → H ∗ ( E, E − Z f ) → · · · . By exactness, there exists λ ∈ H ∗ ( E, E − Z f ) such that j ∗ ( λ ) = W ′ ( x ) | E ,where j : E → ( E, E − Z f )is the natural inclusion. Thus, q ( x, y ) W ′ ( x ) | E = j ∗ ( µ ) ⌣ j ∗ ( λ ) = j ∗ ( µ ⌣ λ ) ARAMETRIZED BORSUK-ULAM PROBLEM 11 by the naturality of the cup product. But, µ ⌣ λ ∈ H ∗ (cid:0) E, V ∪ ( E − Z f ) (cid:1) = H ∗ ( E, E ) = 0and hence q ( x, y ) W ′ ( x ) | E = 0. Therefore, by equation (5.2), there existpolynomials r ( x, y ) and r ( x, y ) in H ∗ ( B )[ x, y ] such that q ( x, y ) W ′ ( x ) = r ( x, y ) W ( x, y ) + r ( x, y ) W ( x, y )in the ring H ∗ ( B )[ x, y ]. This proves the theorem. ✷ Proof of Corollary 4.2.
Let q ( x, y ) in H ∗ ( B )[ x, y ] be a non zero polynomialsuch that deg( q ( x, y )) < ( n − k + 1). If q ( x, y ) | Z f = 0, then by theorem 4.1,we have q ( x, y ) W ′ ( x ) = r ( x, y ) W ( x, y ) + r ( x, y ) W ( x, y )in the ring H ∗ ( B )[ x, y ] for some polynomials r ( x, y ) and r ( x, y ) in H ∗ ( B )[ x, y ].Note that deg( W ′ ( x )) = k , deg( W ( x, y )) = n + 1 and deg( W ( x, y )) = 2.Sincedeg( q ( x, y )) + k = max { deg( r ( x, y )) + n + 1 , deg( r ( x, y )) + 2 } , we have deg( q ( x, y )) + k ≥ deg( r ( x, y )) + n + 1 . Taking deg( r ( x, y )) = 0, this gives deg( q ( x, y )) + k ≥ n + 1 and hencedeg( q ( x, y )) ≥ ( n − k + 1), which is a contradiction. Hence q ( x, y ) | Z f = 0.Equivalently, the H ∗ ( B )-homomorphism n − k M i + j =0 H ∗ ( B ) x i y j → H ∗ ( Z f )given by x i → x i | Z f and y j → y j | Z f is a monomorphism. As a result, if n ≥ k , then cohom.dim ( Z f ) ≥ cohom.dim ( B ) + ( n − k ) , since cohom.dim ( Z f ) ≥ cohom.dim ( Z f ) by [13, Proposition A.11]. ✷ Remark . If B is a point in the above corollary, then for any Z -equivariantmap f : X ≃ R P n → R k , where n ≥ k , we have cohom.dim ( Z f ) ≥ ( n − k ).Next we prove our results for the complex case. Proof of Theorem 4.3.
Let q ( x, y ) in H ∗ ( B )[ x, y ] be a polynomial such that q ( x, y ) | Z f = 0. By similar arguements as used in the proof of theorem 4.1, we conclude that q ( x, y ) W ′ ( x ) | E = 0. Therefore, by equation (5.4), thereexist polynomials r ( x, y ) and r ( x, y ) in H ∗ ( B )[ x, y ] such that q ( x, y ) W ′ ( x ) = r ( x, y ) W ( x, y ) + r ( x, y ) W ( x )in the ring H ∗ ( B )[ x, y ]. This proves the theorem. ✷ Proof of Corollary 4.4.
Let q ( x, y ) in H ∗ ( B )[ x, y ] be a non zero polynomialsuch that deg( q ( x, y )) < (2 n − k + 2). If q ( x, y ) | Z f = 0, then by theorem4.3, we have q ( x, y ) W ′ ( x ) = r ( x, y ) W ( x, y ) + r ( x, y ) W ( x )in the ring H ∗ ( B )[ x, y ] for some polynomials r ( x, y ) and r ( x, y ) in H ∗ ( B )[ x, y ].Note that deg( W ′ ( x )) = k , deg( W ( x, y )) = 2 n + 2 and deg( W ( x )) = 3.Sincedeg( q ( x, y )) + k = max { deg( r ( x, y )) + 2 n + 2 , deg( r ( x, y )) + 3 } , we have deg( q ( x, y )) + k ≥ deg( r ( x, y )) + 2 n + 2 . Taking deg( r ( x, y )) = 0, this gives deg( q ( x, y )) + k ≥ n + 2 and hencedeg( q ( x, y )) ≥ (2 n − k + 2), which is a contradiction. Hence q ( x, y ) | Z f = 0.Equivalently, the H ∗ ( B )-homomorphism n − k +1 M i + j =0 H ∗ ( B ) x i y j → H ∗ ( Z f )given by x i → x i | Z f and y j → y j | Z f is a monomorphism. As a result, if2 n ≥ k , then cohom.dim ( Z f ) ≥ cohom.dim ( B ) + (2 n − k + 1) . ✷ Remark . If B is a point in the above corollary, then for any Z -equivariantmap f : X ≃ C P n → R k , where 2 n ≥ k , we have cohom.dim ( Z f ) ≥ (2 n − k + 1). ARAMETRIZED BORSUK-ULAM PROBLEM 13 Application to Z -coincidence sets Let (
X, E, π, B ) be a fiber bundle with the hypothesis of section 4. Let E ′′ → B be a k -dimensional vector bundle and let f : E → E ′′ be a fiberpreserving map. Here we do not assume that E ′′ has an involution. Even if E ′′ has an involution, f is not assumed to be Z -equivariant. If T : E → E is a generator of the Z -action, then the Z -coincidence set of f is definedas A f = { x ∈ E | f ( x ) = f ( T ( x )) } . Let V = E ′′ ⊕ E ′′ be the Whitney sum of two copies of E ′′ → B . Then Z acts on V by permuting the coordinates. This action has the diagonal D in V as the fixed point set. Note that D is a k -dimensional sub-bundle of V and the orthogonal complement D ⊥ of D is also a k -dimensional sub-bundleof V . Also note that D ⊥ is Z -invariant and has a Z -action which is freeoutside the zero section. Consider the Z -equivariant map f ′ : E → V givenby f ′ ( x ) = (cid:0) f ( x ) , f ( T ( x )) (cid:1) . The linear projection along the diagonal defines a Z -equivariant fiber pre-serving map g : V → D ⊥ such that g ( V − D ) ⊂ D ⊥ −
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