Cohomology of spaces of Hopf equivariant maps of spheres
aa r X i v : . [ m a t h . A T ] F e b COHOMOLOGY OF SPACES OF HOPF EQUIVARIANT MAPSOF SPHERES
V.A. VASSILIEV
Abstract.
For any natural k ≤ n , the rational cohomology ring of the space ofcontinuous maps S k − → S n − (respectively, S k − → S n − ) equivariant underthe Hopf action of the circle (respectively, of the group S of unit quaternions) isnaturally isomorphic to that of the Stiefel manifold V k ( C n ) (respectively, V k ( H n )). Main theorem
The group S of scalars λ ∈ C , | λ | = 1, acts by multiplications on the unitspheres in the complex vector spaces. Denote by E S ( k, n ) the space of continuousmaps f : S k − → S n − equivariant under these actions, i.e. of maps such that f ( λx ) = λf ( x ) for any λ ∈ S ⊂ C , x ∈ S k − ⊂ C k . The Stiefel manifold V k ( C n ) of orthonormal k -frames in C n can be identified with the space of isometriccomplex linear maps C k → C n of Hermitian spaces, and hence be considered as asubset of E S ( k, n ). In a similar way, the quaternionic Stiefel manifold V k ( H n ) canbe considered as a subset of the space E S ( k, n ) of continuous maps S k − → S n − equivariant under the standard action (by left multiplications) of the group S ofquaternions of unit norm. Theorem 1.
For any natural k ≤ n , the maps (1) H ∗ ( E S ( k, n ) , Q ) → H ∗ ( V k ( C n ) , Q ) , (2) H ∗ ( E S ( k, n ) , Q ) → H ∗ ( V k ( H n ) , Q ) induced by inclusions are isomorphisms. Preliminary remarks
Recall (see [1]) that there are ring isomorphisms(3) H ∗ ( V k ( C n ) , Z ) ≃ H ∗ ( S n − × S n − − × · · · × S n − k +1) − , Z ) , (4) H ∗ ( V k ( H n ) , Z ) ≃ H ∗ ( S n − × S n − − × · · · × S n − k +1) − , Z ) . We replace the space E S ( k, n ) by its subspace ˜ E S ( k, n ) of C ∞ -smooth equivari-ant maps (which is homology equivalent to it by a kind of Weierstrass approximation Mathematics Subject Classification.
Key words and phrases.
Equivariant maps, configuration space, Stiefel manifold. theorem, cf. [4]). Using the standard techniques of finite-dimensional approxima-tions (see e.g. [2], [4]) we will work with these spaces of maps as with manifolds ofvery large but finite dimensions.A singular m -dimensional simplex in ˜ E S ( k, n ) is a continuous map F : S k − × ∆ m → S n − , where ∆ m is the standard m -dimensional simplex and all maps F ( · , κ ) : S k − → S n − , κ ∈ ∆ m , are of class ˜ E S ( k, n ). Again by approxima-tion theorem, the chain complex of these simplices has the same homology as itssubcomplex of C ∞ -smooth maps F .3. Surjectivity of the map (1)
Fix an orthonormal frame { e , . . . , e k } in C k and the corresponding complete flag C ⊂ · · · ⊂ C k − ⊂ C k where any C j is spanned by e , . . . , e j . Consider C k as asubspace in C n , so that the source S k − ⊂ C k of our equivariant maps is a subsetof their target S n − ⊂ C n . For any j = 1 , , . . . , k, denote by Ξ j the subset in˜ E S ( k, n ) consisting of all S -equivariant smooth maps S k − → S n − which areidentical on some big circle in S k − cut by a complex line through the origin in C j ⊂ C k . Definition 1. A S -equivariant smooth map f ∈ Ξ j is generic if it is identical onlyon finitely many big circles in S k − ∩ C j , and for any such circle, any point > ofthis circle and any (2 k − S k − , thedifferential of the map f − Id from this slice to C n is injective (i.e. has rank 2 k − > . (A map f can be generic in Ξ j and not generic in some Ξ l , l > j .) Lemma 1.
1. For any j = 1 , . . . , k , the set of generic maps f ∈ Ξ j is a smoothimmersed submanifold of codimension n − j + 1 in the space ˜ E S ( k, n ) in thefollowing exact sense: for any natural m and open domain U ⊂ R m there is aresidual subset in the space of m -parametric smooth families F : S k − × U → S n − of maps of class ˜ E S ( k, n ) , such that for any family from this subset the points κ ∈ U such that F ( · , κ ) ∈ Ξ j form a smooth immersed submanifold of codimension n − j + 1 in U .2. For any generic map f ∈ Ξ j , the local irreducible branches of this immersedmanifold at the point f in ˜ E S ( k, n ) are in a natural one-to-one correspondence withbig circles in C j ∩ S k − on which this map is identical.3. All these local branches have a natural coorientation ( i.e. orientation of thenormal bundle ) in ˜ E S ( k, n ) .4. The set of non-generic maps f ∈ Ξ j has codimension at least (2 n − j + 1) +2 + 2( n − k ) in ˜ E S ( k, n ) .Proof. Items 1 and 4 follow from Thom transversality theorem. The local branch ofΞ j corresponding to a big circle fixed by f , which is assumed in item 2, consists ofmaps which are identical on certain big circles neighboring to this one. QUIVARIANT MAPS OF SPHERES 3
The coorientation assumed in item 3 is induced from the standard orientations ofthe target space S n − and the space C P j − of big circles in C j ∩ S k − . Namely, let f be a generic point of Ξ j and > a point of C j ∩ S k − such that f ( > ) = > . Denoteby Ξ( > ) the local branch of the variety Ξ j ⊂ ˜ E S ( k, n ) at f which corresponds tothe big circle { e it > } containing > . Choose an arbitrary transversal slice L of thisbig circle in C j ∩ S k − . The canonical orientation of L is induced from that of C P j − by the Hopf projection C j ∩ S k − → C P j − . By Definition 1 the differentialof the map f − Id : L → C n is injective at the point > ; obviously it acts to thetangent space of S n − at this point. The normal space in ˜ E S ( k, n ) of the localbranch Ξ( > ) at the point f can be identified with the quotient space of T > ( S n − )by the image of this differential. Indeed, let us realize some tangent vector of thespace ˜ E S ( k, n ) at f by a one-parameter family of maps f τ , τ ∈ ( − ε, ε ), f ≡ f .This vector is tangent to the branch Ξ( > ) if and only if there is a family of points > ( τ ) ∈ L , > (0) = > , such that | f τ ( > ( τ )) − > ( τ ) | = τ → o ( | τ | ) or, which is the same,the vector df τ ( > ) dτ (0) ∈ T > ( S n − ) belongs to the image of the tangent space T > ( L )under the differential of the map f − Id.A canonical orientation of the latter image is defined by that of L , and the desiredorientation of the quotient space (i.e. of the normal space of E ( > ) at f ) is specifiedas the factor of the standard orientation of S n − by this one. (cid:3) Corollary 1.
The intersection index with the set Ξ j defines an element [Ξ j ] of thegroup H n − j +1 ( ˜ E S ( k, n )) for any j = 1 , . . . , k . (cid:3) Proposition 1.
The restrictions of k cohomology classes [Ξ j ] ∈ H n − j )+1 ( ˜ E S ( k, n )) , j = 1 , . . . , k , to the subspace V k ( C n ) ⊂ ˜ E S ( k, n ) generate multiplicatively the ring H ∗ ( V k ( C m ) , Z ) .Proof. For any j = 0 , , . . . , k define the subset A k − j ⊂ V k ( C n ) as the set of isomet-ric linear maps C k → C n sending the first j vectors e , . . . , e j of the basic frame to ie , . . . , ie j respectively. Obviously, any A j is diffeomorphic to V j ( C n − k + j ) , and wehave inclusions { i · Id( C k ) } ≡ A ⊂ A ⊂ A ⊂ · · · ⊂ A k − ⊂ A k ≡ V k ( C n ) , whereany A k − j is embedded into A k − j +1 as a fiber of the fiber bundle A k − j +1 → S n − j )+1 taking a map C k → C n into the image of the basic vector e j . The manifolds A k − j +1 and Ξ j meet transversally, and their intersection is equal to a different fiber of thesame fiber bundle; hence the restriction of the cohomology class [Ξ j ] to A k − j (re-spectively, to A k − j +1 ) is equal to zero (respectively, coincides with the class inducedfrom the basic cohomology class of S n − j )+1 by the projection map of this fiberbundle). Therefore our proposition follows by induction over the spectral sequencesof these fiber bundles (and the multiplicativity of these spectral sequences). (cid:3) Corollary 2.
The map ( ) is epimorphic. (cid:3) V.A. VASSILIEV Injectivity of (1)
For any topological space X and natural number s , denote by B ( X, s ) the s -configuration space of X (i.e., the space of s -element subsets of X ). Denote by ± Z the sign local system of groups on B ( X, s ): it is locally isomorphic to Z , but loopsin B ( X, s ) act on its fibers by multiplications by 1 or − s points defined by these loops. ¯ H ∗ denotes the Borel–Moorehomology group (i.e. the homology group of the complex of locally finite singularchains). Proposition 2.
There is a spectral sequence E p,qr converging to the group ˜ H ∗ ( E S ( k, n ) , Z ) ,whose term E p,q is trivial if p ≥ , and is defined by the formula (5) E p,q ≃ ¯ H − pn − q ( B ( C P k − , − p ) , ± Z ) for p < . The construction of this spectral sequence essentially repeats that of the spectralsequence assumed in Theorem 3 of [4], see also [2]. Namely, we consider the vectorspace of all S -equivariant smooth maps S k − → C n , define the discriminant set init as the set of maps whose images contain the origin, and calculate its “homology offinite codimension” (which is Alexander dual to the usual cohomology group of thespace of equivariant maps to C n \ ∼ S n − ) by using a natural simplicial resolution. (cid:3) Proposition 3 (see [3]) . For any j ≥ and s ≥ there is a group isomorphism (6) ¯ H j ( B ( C P k − , s ) , ± Z ⊗ Q ) ≃ H j − s ( s − ( G s ( C k ) , Q ) , where G s ( C k ) is the Grassmann manifold of s -dimensional complex subspaces in C k .In particular, the left-hand group ( ) is trivial if s > k . (cid:3) Therefore the total dimension of the group ˜ H ∗ ( E S ( k, n ) , Q ) does not exceed k X s =1 dim H ∗ ( G s ( C k ) , Q ) = k X s =1 (cid:18) ks (cid:19) = 2 k − , which by (3) is equal to the total dimension of the reduced homology group of thespace V k ( C n ). So, (1) is an epimorphic (by Corollary 2) map, the dimension of whosesource does not exceed that of the target; hence it is an isomorphism.The bijectivity of the map (2) can be proved in exactly the same way. In particular,the first term of the spectral sequence calculating the group ˜ H ∗ ( E S ( k, n ) , Z ) andanalogous to (5) is given by(7) E p,q ≃ ¯ H − pn − q ( B ( H P k − , − p ) , ± Z )for p < p ≥
0; the quaternionic version of the equality (6) is(8) ¯ H j ( B ( H P k − , s ) , ± Z ⊗ Q ) ≃ H j − s ( s − ( G s ( H k ) , Q ) . QUIVARIANT MAPS OF SPHERES 5
References [1] A. Borel, Sur la cohomologie des espaces fibr´es principaux et des espaces homog`enes de groupesde Lie compacts, Ann. Math., 57 (1953), 115-207.[2] V. A. Vassiliev, Complements of discriminants of smooth maps: topology and applications,Second edition, Amer. Math. Soc., Providence, RI, 1994.[3] V. A. Vassiliev, How to calculate homology groups of spaces of nonsingular algebraic projectivehypersurfaces, Proc. Steklov Inst. Math., 225 (1999), 121–140 arXiv: 1407.7229[4] V. A.Vassiliev, Twisted homology of configuration spaces, homology of spaces of equivariantmaps, and stable homology of spaces of non-resultant systems of real homogeneous polynomials,to appear, arXiv: 1809.05632
Steklov Mathematical Institute of Russian Academy of Sciences and Na-tional Research University Higher School of Economics
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