Free actions of some compact groups on Milnor manifolds
aa r X i v : . [ m a t h . A T ] F e b FREE ACTIONS OF SOME COMPACT GROUPS ON MILNOR MANIFOLDS
PINKA DEY AND MAHENDER SINGH
Abstract.
In this paper, we investigate free actions of some compact groups on cohomology realand complex Milnor manifolds. More precisely, we compute the mod 2 cohomology algebra ofthe orbit space of an arbitrary free Z and S -action on a compact Hausdorff space with mod 2cohomology algebra of a real or a complex Milnor manifold. As applications, we deduce someBorsuk-Ulam type results for equivariant maps between spheres and these spaces. For the complexcase, we obtain a lower bound on the Schwarz genus, which further establishes the existence ofcoincidence points for maps to the Euclidean plane. Introduction
A basic problem in the theory of transformation groups is to determine groups that can actfreely on a given topological space. Once we know that a group acts freely on a given space, thenext natural problem is to determine all actions of the group up to conjugation. Determiningthe homeomorphism or homotopy type of the orbit space is, in general, a difficult problem. Anon-trivial result of Oliver [20] states that the orbit space of any action of a compact Lie groupon a Euclidean space is contractible. For spheres, Milnor [13] proved that for any free involutionon S n , the orbit space has the homotopy type of R P n . Free actions of finite groups on spheres,particularly S , have been well-studied in the past, see for example [23, 24, 26]. But, not manyresults are known for compact manifolds other than spheres. In [19], Myers investigated orbitspaces of free involutions on three-dimensional lens spaces. In [30], Tao determined orbit spaces offree involutions on S × S , and Ritter [25] extended these results to free actions of cyclic groupsof order 2 n . Tollefson [31] proved that there are precisely four conjugacy classes of involutions on S × S . Fairly recently, Jahren and Kwasik [14] classified, up to conjugation, all free involutions on S × S n for n ≥
3, by showing that there are exactly four possible homotopy types of orbit spaces.Various attempts have been made towards the weaker problem of determining possible cohomol-ogy algebra of orbit spaces of free actions of finite groups on some specific classes of manifolds, forexample, products of spheres, spherical space forms and their products. Dotzel et al. [10] deter-mined the cohomology algebra of orbit spaces of free Z p ( p prime) and S -actions on cohomologyproduct of two spheres. Orbit spaces of free involutions on cohomology lens spaces were investi-gated by Singh [28]. The cohomology algebra of orbit spaces of free involutions on product of twoprojective spaces was computed in another paper [27]. Recently, Pergher et al. [21] and Mattoset al. [9] considered free Z and S -actions on spaces of type ( a, b ), which are certain products orwedge sums of spheres and projective spaces. As applications, they also established some bundletheoretic analogues of Borsuk-Ulam theorem for these spaces.Viewing the product of two projective spaces as a trivial bundle, it is interesting to consider non-trivial projective space bundles over projective spaces. Milnor manifolds are fundamental examples Mathematics Subject Classification.
Primary 57S25, 57S10; Secondary 55R20, 55M20.
Key words and phrases.
Borsuk-Ulam theorem; cohomology algebra; equivariant map; index of involution; Leray-Serre spectral sequence; Milnor manifold; Schwarz genus. of such spaces. It is well-known that the unoriented cobordism algebra of smooth manifolds isgenerated by the cobordism classes of real projective spaces and real Milnor manifolds [17, Lemma1]. Therefore, determining various invariants of these manifolds is of interest. Free actions ofelementary abelian 2-groups on products of Milnor manifolds have been investigated in [29], whereinsome bounds on the rank of these groups are determined.The aim of this paper is to investigate free Z and S -actions on mod 2 cohomology real andcomplex Milnor manifolds. More precisely, we determine the possible mod 2 cohomology algebraof orbit spaces of free Z and S -actions on these spaces. We also find necessary and sufficientconditions for the existence of free actions on these spaces. As applications, we obtain someBorsuk-Ulam kind results for these spaces. We also determine some lower bound on the genus andresults on the existence of non-empty coincidence set.The paper is organized as follows. In Section 2, we recall the definition and cohomology ofMilnor manifolds. In Section 3, we construct free Z n and S -actions on these manifolds. Section 4consists of some preliminaries from the theory of compact transformation groups that will be usedin subsequent sections. Induced action on cohomology is investigated in Section 5. In Section 6,we prove our main results as Theorems 6.1, 6.2 and 6.3. Finally, in Section 7, we deduce someBorsuk-Ulam type results for equivariant maps between spheres and these spaces. For the complexcase, we obtain a lower bound on the Schwarz genus, which establishes the existence of coincidencepoints for maps to the Euclidean plane.2. Milnor manifolds
Let r and s be integers such that 0 ≤ s ≤ r . A real Milnor manifold, denoted by R H r,s , isthe non-singular hypersurface of degree (1 ,
1) in the product R P r × R P s of real projective spaces.Milnor [17] introduced these manifolds in search for generators for the unoriented cobordism algebra.Clearly, R H r,s is a ( s + r − R H r,s = n(cid:0) [ x , . . . , x r ] , [ y , . . . , y s ] (cid:1) ∈ R P r × R P s | x y + · · · + x s y s = 0 o . Alternatively, R H r,s is given as the total space of the fiber bundle R P r − i ֒ → R H r,s p −→ R P s . This is projectivization of the vector bundle R r ֒ → E ⊥ −→ R P s , where E ⊥ is the orthogonal complement in R P s × R r +1 of the canonical line bundle R ֒ → E −→ R P s . Similarly, a complex Milnor manifold, denoted by C H r,s , is a 2( s + r − C H r,s = n(cid:0) [ z , . . . , z r ] , [ w , . . . , w s ] (cid:1) ∈ C P r × C P s | z w + · · · + z s w s = 0 o . As in the real case, C H r,s is the total space of the fiber bundle C P r − i ֒ → C H r,s p −→ C P s . It is known due to Conner and Floyd [8, p.63] that C H r,s is unoriented cobordant to R H r,s × R H r,s . ROUP ACTIONS ON MILNOR MANIFOLDS 3
These manifolds have been well-studied in the past. See, [11, 15, 29] for some recent results.Their cohomology algebra is also well-known [5, 18], and we need it for the proofs of our mainresults.
Theorem 2.1.
Let ≤ s ≤ r . Then the following holds: (1) H ∗ ( R H r,s ; Z ) ∼ = Z [ a, b ] / h a s +1 , b r + ab r − + · · · + a s b r − s i ,where a and b are homogeneous elements of degree one each. (2) H ∗ ( C H r,s ; Z ) ∼ = Z [ g, h ] / h g s +1 , h r + gh r − + · · · + g s h r − s i ,where g and h are homogeneous elements of degree two each. Note that, R H r, = R P r − and C H r, = C P r − . Since orbit spaces of free involutions on realand complex projective spaces are well-known, we, henceforth, assume that 1 ≤ s ≤ r .3. Free actions on Milnor manifolds
Circle Actions.
We give examples of free S -actions on R H r,s in the case when both r and s are odd, and later prove that this is indeed a necessary condition for the existence of a free S -action. We first give a free S -action on R P s . Note that, only odd-dimensional real projectivespaces admit free S -actions. Let s = 2 m + 1 and write an element of R P m +1 as [ w , . . . , w m ],where w i are complex numbers.Define a map S × R P m +1 → R P m +1 by (cid:0) ξ, [ w , . . . , w m ] (cid:1) = [ p ξw , . . . , p ξw m ] . It can be checked that the preceding map gives a free S -action on R P m +1 .Let r = 2 n + 1 and write an element of R P r as [ z , . . . , z n ], where z i are complex numbers. Definean action of S on R P r by (cid:0) ξ, [ z , . . . , z n ] (cid:1) = [ p ξz , . . . , p ξz n ] . The diagonal action on R P r × R P s is free, and R H r,s is invariant under this action giving rise to afree S -action on R H r,s . Restricting the above S -action gives free Z n (in particular Z ) action on R H r,s .If X and Y are two spaces, then X ∼ = Y means that X and Y have isomorphic mod 2 cohomologyalgebras, not necessarily induced by a map between X and Y . For the complex case, we have thefollowing result. Proposition 3.1.
There is no free S -action on a compact Hausdorff space X ≃ C H r,s .Proof. Recall that, we have a fiber bundle C P r − ֒ → C H r,s −→ C P s with χ ( C H r,s ) = r ( s + 1). Suppose there is a free S -action on X . Restriction of this action givesfree Z p -actions for each prime p . By Floyd’s Euler characteristic formula [4, Chapter III, Theorem7.10], we have r ( s + 1) = χ ( X ) = p χ ( X/ Z p )for each prime p , which is a contradiction. Hence, there is no free S -action on a space X ≃ C H r,s . (cid:3) PINKA DEY AND MAHENDER SINGH
Involutions. If s = r , then interchanging the coordinates i.e., (cid:0) [ z , . . . , z s ] , [ w , . . . , w s ] (cid:1) (cid:0) [ w , . . . , w s ] , [ z , . . . , z s ] (cid:1) gives a free involution on Milnor manifolds. But, if 1 < s < r and r R H r,s (respectively C H r,s ) admits a free involution if and only if both r and s are odd. Wehave seen examples of free involutions on real Milnor manifolds before.For the complex case, it is known that C P n admits a free action by a finite group if and only if n is odd, and in that case the only possible group is Z [12].If s is odd, then the map[ z , z , . . . , z s − , z s ] [ − z , z , . . . , − z s , z s − ] , defines a free involution on C P s . Similarly, for r odd, the map[ z , z , . . . , z r − , z r ] [ − z , z , . . . , − z r , z r − ] , is a free involution on C P r . Hence, the diagonal action on C P r × C P s is free and its restrictiongives a free involution on C H r,s . 4. Preliminaries
For the convenience of the reader, we recall some facts that we use without mentioning explicitly.For further details, we refer the reader to [1, 4, 16]. Throughout, we use cohomology with Z coefficients, and suppress it from the notation.Let G be a group and X a G -space. Let G ֒ → E G −→ B G be the universal principal G -bundle and X i ֒ → X G π −→ B G the associated Borel fibration [3, Chapter IV]. Our main computational tool is the Leray-Serrespectral sequence associated to the Borel fibration [16, Theorem 5.2]. The E -term of this spectralsequence is given by E k,l = H k (cid:0) B G ; H l ( X ) (cid:1) , where H l ( X ) is a locally constant sheaf with stalk H l ( X ) and group G . Further, the spectralsequence converges to H ∗ ( X G ) as an algebra. If π ( B G ) acts trivially on H ∗ ( X ), then the systemof local coefficient is simple and we get E k,l ∼ = H k ( B G ) ⊗ H l ( X ) . Further, if the system of local coefficient is simple, then the edge homomorphisms H k ( B G ) = E k, −→ E k, −→ · · · −→ E k, k −→ E k, k +1 = E k, ∞ ⊂ H k ( X G )and H l ( X G ) −→ E ,l ∞ = E ,ll +1 ⊂ E ,ll ⊂ · · · ⊂ E ,l = H l ( X )are the homomorphisms π ∗ : H k ( B G ) → H k ( X G ) and i ∗ : H l ( X G ) → H l ( X ) , respectively [16, Theorem 5.9]. ROUP ACTIONS ON MILNOR MANIFOLDS 5
On passing to quotients, the G -equivariant projection X × E G → X yields the fiber bundle E G ֒ → X G h −→ X/G with contractible fiber E G . By [1, p. 20], h is a homotopy equivalence, and consequently h ∗ : H ∗ ( X/G ) ∼ = −→ H ∗ ( X G ) . Next, we recall some results regarding free Z and S -actions on compact Hausdorff spaces. Forfree actions, vanishing of H ∗ ( X ) implies vanishing of H ∗ ( X/G ) in higher range [4, p. 374, Theorem1.5].
Proposition 4.1.
Let G = Z act freely on a compact Hausdorff space X . Suppose that H j ( X ) = 0 for all j > n , then H j ( X/G ) = 0 for all j > n . For G = S , one can derive an analogue of the preceding result by using the Gysin-sequence forthe principle bundle X → X/G . Proposition 4.2.
Let G = S act freely on a compact Hausdorff space X . Suppose that H j ( X ) = 0 for all j > n , then H j ( X/G ) = 0 for all j ≥ n . We use the wellknown facts that H ∗ ( B Z ; Z ) = Z [ t ] and H ∗ ( B S ; Z ) = Z [ u ], where deg( t ) = 1and deg( u ) = 2, respectively. 5. Induced action on cohomology
When a group acts on a topological space, in general, it is difficult to determine the inducedaction on cohomology. In our context, we have the following
Proposition 5.1.
Let G = Z act freely on a compact Hausdorff space X ≃ R H r,s , where
0. Therefore, it is enough to consider g ∗ : H ( X ) → H ( X ) . Suppose that g ∗ is non-trivial. Then it cannot preserve both a and b . Assuming that g ∗ ( b ) = b ,we have g ∗ ( b ) = a or a + b . If g ∗ ( b ) = a , then g ∗ ( b s +1 ) = a s +1 = 0, which implies b s +1 = 0.Hence, b r = 0, contradicting the fact that top dimensional cohomology must be non-zero with Z coefficients. So, we must have g ∗ ( b ) = a + b and g ∗ ( a ) = a . Suppose that r is odd. Then g ∗ ( a s − b r ) = a s − ( a + b ) r = ra s b r − + a s − b r = a s b r − + a s − b r = 0 . This gives a s − b r = 0, which is a contradiction. Hence, for r odd, the induced action on H ∗ ( X )must be trivial.Suppose that r ≡ b r +1 = 0 implies g ∗ ( b r +1 ) = ( a + b ) r +1 = 0. But, fromthe binomial expansion( a + b ) r +1 = a r +1 + · · · + (cid:18) r + 12 (cid:19) a b r − + ( r + 1) ab r , PINKA DEY AND MAHENDER SINGH we see that the last term is non-zero and the second last term is zero modulo 2. This gives( a + b ) r +1 = 0, a contradiction. Hence, the induced action must be trivial in this case as well. (cid:3) Remark . If s = 1 and r > b and a + b are r + 1 and r ,respectively. Hence, in this case also g ∗ is identity. For s = 1 or r ≡ H ∗ ( X ) may be non-trivial.Similarly, for the complex case, we have the following Proposition 5.3.
Let G = Z act freely on a compact Hausdorff space X ≃ C H r,s , where
Let G = Z act freely on X ≃ R H r,s , where < s < r and r .Then both r and s are odd.Proof. Suppose Z acts freely on X ≃ R H r,s . Let a , b ∈ H ( X ) be generators of the cohomologyalgebra H ∗ ( X ). By Proposition 5.1, π ( B G ) = Z acts trivially on H ∗ ( X ), so that the fibration X ֒ → X G −→ B G has a simple system of local coefficients. Hence the spectral sequence has theform E p,q ∼ = H p ( B G ) ⊗ H q ( X ) . If d : E , → E , is trivial, then the spectral sequence degenerates at E -term and we get H i ( X/G ) = 0 for infinitely many values of i . This contradicts Proposition 4.1. Thus d must benon-trivial. Hence we have following three possibilities:(i) d (1 ⊗ a ) = t ⊗ d (1 ⊗ b ) = 0.(ii) d (1 ⊗ a ) = 0 and d (1 ⊗ b ) = t ⊗ d (1 ⊗ a ) = t ⊗ d (1 ⊗ b ) = t ⊗ s is odd, first we show that case (i) is not possible. The even case follows similarly.Suppose d (1 ⊗ a ) = t ⊗ d (1 ⊗ b ) = 0. By the derivation property of the differential, wehave d ( t k ⊗ a m b n ) = (cid:26) t k +2 ⊗ a m − b n if m is odd0 if m is even. ROUP ACTIONS ON MILNOR MANIFOLDS 7
Then the relation b r + ab r − + · · · + a s b r − s = 0 gives0 = d (1 ⊗ ( b r + ab r − + · · · + a s b r − s ))= d (1 ⊗ b r ) + d (1 ⊗ ab r − ) + · · · + d (1 ⊗ a s b r − s )= 0 + t ⊗ b r − + · · · + t ⊗ a s − b r − s = t ⊗ ( b r − + · · · + a s − b r − s ) , a contradiction. Hence case (i) is not possible. The same argument works for case (ii) as well.Hence, we must have d (1 ⊗ a ) = t ⊗ d (1 ⊗ b ) = t ⊗
1. If s is even, then a s +1 = 0 gives0 = d (1 ⊗ a s +1 ) = t ⊗ a s , a contradiction. Therefore, s must be odd, and a similar argument shows that r is also odd. (cid:3) As a consequence of Proposition 5.5 and the previously defined S -action on R H r,s , we obtainthe following Corollary 5.6.
Let < s < r and r . Then R H r,s admits a free involution if andonly if both r and s are odd. We have similar observations for the complex case.
Proposition 5.7.
Let Z act freely on X ≃ C H r,s with < s < r and r . Then both r and s are odd. Corollary 5.8.
Let < s < r and r . Then C H r,s admits a free involution if andonly if both r and s are odd. For S -actions, we have the following Proposition 5.9.
Let ≤ s ≤ r . Then S acts freely on R H r,s if and only if both r and s are odd.Proof. For G = S , since π ( B G ) = 1, the system of local coefficients is simple. Recall that, H ∗ ( B S ; Z ) = Z [ u ], where deg( u ) = 2. Hence, the spectral sequence has the form E p,q ∼ = H p ( B G ) ⊗ H q ( X ) . Clearly, for p odd, E p,q = 0. As in the case of Z -action, it can be seen that the differential d mustbe non-zero and the only possibility for d is d (1 ⊗ a ) = u ⊗ d (1 ⊗ b ) = u ⊗
1. Consequently,both r and s must be odd. (cid:3) Main results
We are now in a position to present our main results.
Theorem 6.1.
Let G = Z act freely on a compact Hausdorff space X ≃ R H r,s such that inducedaction on mod 2 cohomology is trivial. Then H ∗ ( X/G ; Z ) ∼ = Z [ x, y, z, w ] /I, where I = D z , w − γ zw − γ x − γ y, x s +12 + α zwx s − + α zwx s − y + · · · + α s − zwy s − , PINKA DEY AND MAHENDER SINGH ( w + β z ) y r − + ( w + β z ) xy r − + · · · + ( w + β s − z ) x s − y r − s E , with deg( x ) = 2 , deg( y ) = 2 , deg( z ) = 1 , deg( w ) = 1 and α i , β i , γ i ∈ Z .Proof. Let a, b ∈ H ( X ) be generators of the cohomology algebra H ∗ ( X ). By similar argument asin Proposition 5.5, we see that both r and s must be odd and d (1 ⊗ a ) = t ⊗ d (1 ⊗ b ) = t ⊗ . By the derivation property of the differential, we have d (1 ⊗ a m b n ) = t ⊗ a m − b n + t ⊗ a m b n − if m and n are odd t ⊗ a m − b n if m is odd and n is even t ⊗ a m b n − if m is even and n is odd0 if m and n are even.It suffices to look at d : E ,q → E ,q − . • For q ≤ s , a basis of E ,q ∼ = H q ( X ) consists of { a q , a q − b, . . . , ab q − , b q } . If q is even, then rk(Ker d ) = q + 1 and rk(Im d ) = q . If q is odd, then rk(Ker d ) = q +12 =rk(Im d ). • For s < q ≤ r −
1, a basis consists of { a s b q − s , a s − b q − s +1 , . . . , ab q − , b q } . In this case, rk(Ker d ) = s +12 = rk(Im d ). • For r ≤ q ≤ s + r −
1, a basis consists of { a s b q − s , a s − b q − s +1 , . . . , a q − r +1 b r − } . If q is odd, then rk(Ker d ) = s + r − − q and rk(Im d ) = s + r +1 − q . And, if q is even, then rk(Ker d ) = r + s − q = rk(Im d ).From the above observation, we get that for all k ≥ l , E k,l = 0 as rk( E k,l ) = 0.This gives E k,l = (cid:26) Ker { d : E k,l → E k +2 ,l − } k = 0, 1 and for all l .0 k ≥ l .Note that d r : E k,lr → E k + r,l − r +1 r is trivial for all r ≥ k, l , and hence E ∗ , ∗∞ ∼ = E ∗ , ∗ . Since H ∗ ( X G ) ∼ = Tot E ∗ , ∗∞ , the total complex of E ∗ , ∗∞ , we have H n ( X G ) ∼ = M i + j = n E i,j ∞ = E ,n ∞ ⊕ E ,n − ∞ for all 0 ≤ n ≤ r + s − t ⊗ z = π ∗ ( t ) ∈ E , ∞ ⊆ H ( X G ) be determined by t ⊗ ∈ E , . As E , ∞ = 0, we have z = 0. Also, 1 ⊗ ( a + b ) ∈ E , is a permanent cocycle. Let w ∈ H ( X G ) such that i ∗ ( w ) = a + b . Notice that, 1 ⊗ a ∈ E , and 1 ⊗ b ∈ E , are permanentcocycles, and hence they determine elements in E , ∞ . Let x, y ∈ H ( X G ) such that i ∗ ( x ) = a and i ∗ ( y ) = b . As a s +1 = 0 , we get the following relation x s +12 + α zwx s − + α zwx s − y + · · · + α s − zwy s − = 0 , ROUP ACTIONS ON MILNOR MANIFOLDS 9 where α i ∈ Z . Notice that, i ∗ ( wy r − + wxy r − + · · · + wx s − y r − s ) = 0 . Hence it satisfies wy r − + wxy r − + · · · + wx s − y r − s = β zy r − + β zxy r − + β s − zx s − y r − s , where β i ∈ Z . Note that, we can write w as the following w = γ zw + γ x + γ y, where γ i ∈ Z . Therefore H ∗ ( X/G ) ∼ = H ∗ ( X G ) ∼ = Z [ x, y, z, w ] /I, where I = D z , w − γ zw − γ x − γ y, x s +12 + α zwx s − + α zwx s − y + · · · + α s − zwy s − , ( w + β z ) y r − + ( w + β z ) xy r − + · · · + ( w + β s − z ) x s − y r − s E , with deg( x ) = 2, deg( y ) = 2, deg( z ) = 1, deg( w ) = 1 and α i , β i , γ i ∈ Z . (cid:3) For the complex case, we prove the following
Theorem 6.2.
Let G = Z act freely on a compact Hausdorff space X ≃ C H r,s , such that inducedaction on mod 2 cohomology is trivial. Then H ∗ ( X/G ; Z ) ∼ = Z [ x, y, z, w ] /J, where J = D z , w − γ z w − γ x − γ y, x s +12 + α z wx s − + α z wx s − y + · · · + α s − z wy s − , ( w + β z ) y r − + ( w + β z ) xy r − + · · · + ( w + β s − z ) x s − y r − s E , with deg( x ) = 4 , deg( y ) = 4 , deg( z ) = 1 , deg( w ) = 2 and α i , β i , γ i ∈ Z .Proof. Let G = Z act freely on X ≃ C H r,s . Note that E k,l = 0 for l odd. This gives d : E k,l → E k +2 ,l − is zero, and hence E k,l = E k,l for all k , l . Let a , b ∈ H ( X ) be generators of the cohomologyalgebra H ∗ ( X ). As in the proof of the Theorem 6.1, the only possibility for d is d (1 ⊗ a ) = t ⊗ d (1 ⊗ b ) = t ⊗ . Note that r and s must be odd. For various values of l , we consider the differentials d : E , l → E , l − . If we compute the ranks of Ker d and Im d , we get that rk( E k, l ) = 0 for all k ≥
3. This impliesthat E k, l = 0 for all k ≥ E k, l = Ker { d : E k, l → E k +3 , l − } for k = 0, 1, 2. Also, d r : E k,lr → E k + r,l − r +1 r is zero for all r ≥
4. Hence E ∗ , ∗∞ ∼ = E ∗ , ∗ . Since H ∗ ( X G ) ∼ = Tot E ∗ , ∗∞ , we get H n ( X G ) ∼ = M i + j = n E i,j ∞ = E ,n ∞ ⊕ E ,n − ∞ ⊕ E ,n − ∞ for all 0 ≤ p ≤ s + r − t ⊗ z = π ∗ ( t ) ∈ E , ∞ ⊆ H ( X G ) be determined by t ⊗ ∈ E , . As E , ∞ = 0, we have z = 0. Also, 1 ⊗ ( a + b ) ∈ E , is a permanent cocycle.Let w ∈ H ( X G ) such that i ∗ ( w ) = a + b . Also, 1 ⊗ a ∈ E , and 1 ⊗ b ∈ E , are permanentcocycles, and hence they determine elements in E , ∞ . Let x, y ∈ H ( X G ) such that i ∗ ( x ) = a and i ∗ ( y ) = b . As a s +1 = 0, we get the following relation x s +12 + α z wx s − + α z wx s − y + · · · + α s − z wy s − = 0 , where α i ∈ Z . Notice that, i ∗ ( wy r − + wxy r − + · · · + wx s − y r − s ) = 0 . Hence it satisfies wy r − + wxy r − + · · · + wx s − y r − s = β z y r − + β z xy r − + β s − z x s − y r − s , where β i ∈ Z . Note that, w satisfies the following relation w = γ z w + γ x + γ y, where γ i ∈ Z . Therefore H ∗ ( X/G ) ∼ = H ∗ ( X G ) ∼ = Z [ x, y, z, w ] /J, where J = D z , w − γ z w − γ x − γ y, x s +12 + α z wx s − + · · · + α s − z wy s − , ( w + β z ) y r − + ( w + β z ) xy r − + · · · + ( w + β s − z ) x s − y r − s E , with deg( x ) = 4, deg( y ) = 4, deg( z ) = 1, deg( w ) = 2 and α i , β i , γ i ∈ Z . This completes theproof. (cid:3) For S actions, we obtain the following Theorem 6.3.
Let G = S act freely on a compact Hausdorff space X ≃ R H r,s . Then H ∗ ( X/G ; Z ) ∼ = Z [ x, y, w ] / h x s +12 , wy r − + xwy r − + · · · + wx s − y r − s , w − αx − βy i , where deg( x ) = 2 , deg( y ) = 2 , deg( w ) = 1 and α, β ∈ Z .Proof. By Proposition 5.9, the only possibility for the differential d is that d (1 ⊗ a ) = u ⊗ d (1 ⊗ b ) = u ⊗ r and s are odd. As in the proof of Theorem 6.1, if we compute theranks of Ker d and Im d , we get rk( E k,l ) = 0 and hence E k,l = 0 for all k ≥ l . Also, E ,l = Ker { d : E ,l → E ,l − } for all l .Note that, d r : E k,lr → E k + r,l − r +1 r is trivial for all r ≥ k, l . Hence E ∗ , ∗∞ = E ∗ , ∗ . Since H ∗ ( X G ) ∼ = Tot E ∗ , ∗∞ , we have H n ( X G ) ∼ = M i + j = n E i,j ∞ = E ,n ∞ ROUP ACTIONS ON MILNOR MANIFOLDS 11 for all 0 ≤ n ≤ r + s − ⊗ ( a + b ) ∈ E , is a permanent cocycle. Let w ∈ H ( X G ) such that i ∗ ( w ) = a + b .Also, 1 ⊗ a ∈ E , and 1 ⊗ b ∈ E , are permanent cocycles. Hence they determine elements in E , ∞ . Let x, y ∈ H ( X G ) such that i ∗ ( x ) = a and i ∗ ( y ) = b . As a s +1 = 0, we get x s +12 = 0. Notethat i ∗ ( wy r − + xwy r − + · · · + wx s − y r − s ) = 0 . Hence we get the following relation wy r − + xwy r − + · · · + wx s − y r − s = 0 . Note that, we can write w as the following w = αx + βy, for some α, β ∈ Z . Therefore H ∗ ( X/G ) ∼ = H ∗ ( X G ) ∼ = Z [ x, y, w ] / h x s +12 , wy r − + xwy r − + · · · + wx s − y r − s , w − αx − βy i , where deg( x ) = 2, deg( y ) = 2, deg( w ) = 1 and α, β ∈ Z . (cid:3) Example 6.4.
Take r = 3 and s = 1. Recall that, R H , is a 3-dimensional closed smoothmanifold. A free S -action on R H , gives a principal S -bundle R H , → R H , / S with compact2-dimensional base. Now, using the Leray-Serre spectral sequence associated to the Borel fibration,one can see that H (cid:0) R H , / S ; Z (cid:1) ∼ = Z . Hence, the orbit space must be R P and its cohomologyalgebra matches with our result for β = 1. Corollary 6.5.
Let G = S act freely on a compact Hausdorff space X ≃ R H r,s . Then the Eulerclass of the principal G -bundle X q −→ X/G is zero.Proof.
From Theorem 6.3, we get H i ( X/G ) = Z for i = 0 , H ( X/G ) = Z ⊕ Z . The Gysinsequence of the G -bundle X q −→ X/G is0 −→ H ( X/G ) q ∗ −→ H ( X ) −→ H ( X/G ) ` e −→ H ( X/G ) q ∗ −→ · · · , where e ∈ H ( X/G ) is the Euler class. The conclusion now follows from the sequence. (cid:3) Applications to equivariant maps
Let X be a compact Hausdorff space with a free involution and S n the unit n -sphere equippedwith the antipodal involution. Conner and Floyd [7] asked; for which integer n , there exists a Z -equivariant map from S n to X , but no such map from S n +1 to X .For X = S n , by the Borsuk-Ulam theorem, the answer to the preceding question is n . In thesame paper, Conner and Floyd defined the index of the involution on X asind( X ) = max { n | there exists a Z -equivariant map S n → X } . The characteristic classes with Z coefficients can be used to derive a cohomological criteria tostudy the above question. Let w ∈ H ( X/G ; Z ) be the Stiefel-Whitney class of the principal G -bundle X → X/G . Conner and Floyd also definedco-ind Z ( X ) = max { n | w n = 0 } . Since co-ind Z ( S n ) = n , by [7, (4.5)], we obtainind( X ) ≤ co-ind Z ( X ) . Using these indices, we obtain the following results.
Proposition 7.1.
Let X ≃ R H r,s be a compact Hausdorff space, where ≤ s < r . Then there isno Z -equivariant map S k → X for k ≥ .Proof. Take a classifying map f : X/G → B G for the principal G -bundle X → X/G . Let η : X/G → X G is a homotopy inverse of the homotopyequivalence h : X G → X/G . Then πη : X/G → B G also classifies the principal G -bundle X → X/G ,and hence it is homotopic to f . Therefore it suffices to consider the map π ∗ : H ( B G ) → H ( X G ) . The image of the Stiefel-Whitney class of the universal principal G -bundle G ֒ → E G −→ B G is theStiefel-Whitney class of X → X/G . For X ≃ R H r,s , using the proof of Theorem 6.1, we see that x ∈ H ( X/G ) is the Stiefel-Whitney class with x = 0 and x = 0. This gives co-ind Z ( X ) = 1 andind( X ) ≤
1. Hence, there is no Z -equivariant map S k → X for k ≥ (cid:3) Proposition 7.2.
Let X ≃ C H r,s be a compact Hausdorff space, where ≤ s < r . Then there isno Z -equivariant map S k → X for k ≥ .Proof. From the proof of Theorem 6.2, x ∈ H ( X/G ) is the Stiefel-Whitney class with x = 0and x = 0. This gives co-ind Z ( X ) = 2 and ind( X ) ≤
2. Hence, there is no Z -equivariant map S k → X for k ≥ (cid:3) Given a G -space X , Volovikov [33] defined another numerical index i ( X ) as the smallest r suchthat for some k , the differential d r : E k − r,r − r → E k, r in the Leray-Serre spectral sequence of the fibration X i ֒ → X G π −→ B G is non-trivial. It is clearthat i ( X ) = r if E k, = E k, = · · · = E k, r for all k and E k, r = E k, r +1 for some k . If E ∗ , = E ∗ , ∞ , then i ( X ) = ∞ . Thus, i ( X ) is either an integer greater than 1 or ∞ . Using this index, Coelho,Mattos and Santos proved the following [6, Theorem 1.1] result. Proposition 7.3.
Let G be a compact Lie group and X , Y be path-connected compact Hausdorffspaces with free G -actions. Suppose that i ( X ) ≥ m +1 for some natural m ≥
1. If H k +1 ( Y /G ; Z ) =0 for some ≤ k < m and < rk (cid:0) H k +1 ( B G ) (cid:1) , then there is no G-equivariant map f : X → Y . The preceding result together yields the following
Proposition 7.4.
Suppose Z acts freely on X ≃ C H r,s and a path-connected compact Hausdorffspace Y such that H ( Y /G ) = 0 . Then there is no Z -equivariant map X → Y. Proof.
Note that, we obtained i ( X ) = 3 in the proof of Theorem 6.2. Now the result is a consequenceof Proposition 7.3. (cid:3) ROUP ACTIONS ON MILNOR MANIFOLDS 13
Let G be a finite group considered as a 0-dimensional simplicial complex and X a paracompactspace with a free G -action. The Schwarz genus g free ( X, G ) of the free G -space X is the smallestnumber n such that there exists a G -equivariant map X → G ∗ · · · ∗ G, the n -fold join of G equipped with the diagonal G -action. Note that for G = Z , the free genus isthe least integer n for which there exists a Z -equivariant map f : X → S n − . See [32, Chapter V]for the original source and [2, 33] for more details and applications. In the literature, the free genusfor G = Z is known under different names, for example, B -index [34], co-index [7], level [22]. Proposition 7.5.
Let X ≃ C H r,s be a compact Hausdorff space with a free Z -action. Theng free ( X, Z ) ≥ . In particular, there does not exist any Z -equivariant map X → S .Proof. It follows from [7] that g free ( X, Z ) ≥ co-ind Z ( X ) + 1 , and hence g free ( X, Z ) ≥ (cid:3) Let G be a finite group and X a G -space. Given a continuous map f : X → Y , the coincidenceset A ( f, k ) is defined as A ( f, k ) = { x ∈ X | ∃ distinct g , . . . , g k ∈ G such that f ( g x ) = · · · = f ( g k x ) } . The following result of Schwarz [32] relates the free genus and the coincidence set.
Theorem 7.6.
Let X be a paracompact connected space with a free Z p -action. Suppose that g free ( X, Z p ) > m ( p − . Then for any continuous map f : X → R m g free (cid:0) A ( f, p ) , Z p (cid:1) ≥ g free ( X, Z p ) − m ( p − . In particular, the set A ( f, p ) is non-empty. As a consequence of the preceding theorem and Proposition 7.5, we obtain the following
Proposition 7.7.
Let X ≃ C H r,s be a compact Hausdorff space with a free Z -action. Then anycontinuous map X → R has a non-empty coincidence set.Acknowledgement. Dey thanks UGC-CSIR for the Senior Research Fellowship towards this workand Singh acknowledges support from SERB MATRICS Grant MTR/2017/000018.
References [1] C. Allday and V. Puppe,
Cohomological methods in transformation groups , volume 32 of Cambridge Studies inAdvanced Mathematics. Cambridge University Press, Cambridge, 1993.[2] Thomas Bartsch,
Topological methods for variational problems with symmetries , volume 1560 of Lecture Notesin Mathematics. Springer-Verlag, Berlin, 1993.[3] Armand Borel,
Seminar on transformation groups . With contributions by G. Bredon, E. E. Floyd, D. Mont-gomery, R. Palais, Annals of Mathematics Studies, No. 46. Princeton University Press, Princeton, N.J., 1960.[4] Glen E. Bredon,
Introduction to compact transformation groups , Academic Press, New York-London, 1972. Pureand Applied Mathematics, Vol. 46.[5] V. M. Bukhshtaber and N. Ra˘ı,
Toric manifolds and complex cobordisms , Uspekhi Mat. Nauk, 53(2(320)) : 139 -140, 1998. [6] Francielle R. C. Coelho, Denise de Mattos, and Edivaldo L. dos Santos,
On the existence of G -equivariant maps ,Bull. Braz. Math. Soc. (N.S.), 43(3) : 407 - 421, 2012.[7] P. E. Conner and E. E. Floyd, Fixed point free involutions and equivariant maps , Bull. Amer. Math. Soc.,66 : 416 - 441, 1960.[8] P. E. Conner and E. E. Floyd,
Differentiable periodic maps , Ergebnisse der Mathematik und ihrer Grenzgebiete,N. F., Band 33. Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin-G¨ottingen-Heidelberg, 1964.[9] Denise de Mattos, Pedro Luiz Q. Pergher, and Edivaldo L. dos Santos,
Borsuk-Ulam theorems and theirparametrized versions for spaces of type ( a, b ), Algebr. Geom. Topol., 13(5) : 2827 - 2843, 2013.[10] Ronald M. Dotzel, Tej B. Singh, and Satya P. Tripathi, The cohomology rings of the orbit spaces of free trans-formation groups of the product of two spheres , Proc. Amer. Math. Soc., 129(3) : 921 - 930, 2001.[11] Imma G´alvez and Andrew Tonks,
Differential operators and the Witten genus for projective spaces and Milnormanifolds , Math. Proc. Cambridge Philos. Soc., 135(1) : 123 - 131, 2003.[12] Allen Hatcher,
Algebraic topology , Cambridge University Press, Cambridge, 2002.[13] Morris W. Hirsch and John Milnor,
Some curious involutions of spheres , Bull. Amer. Math. Soc., 70 : 372 - 377,1964.[14] Bjørn Jahren and S lawomir Kwasik,
Free involutions on S × S n , Math. Ann., 351(2) : 281 - 303, 2011.[15] Masayoshi Kamata and Kiyonobu Ono, On the multiple points of the self-transverse immersions of the realprojective space and the Milnor manifold , Kyushu J. Math., 60(2) : 331 - 344, 2006.[16] John McCleary,
A user’s guide to spectral sequences , volume 58 of Cambridge Studies in Advanced Mathematics,Cambridge University Press, Cambridge, second edition, 2001.[17] J. Milnor,
On the Stiefel-Whitney numbers of complex manifolds and of spin manifolds , Topology, 3 : 223 - 230,1965.[18] Himadri Kumar Mukerjee,
Classification of homotopy real Milnor manifolds , Topology Appl., 139(1-3) : 151 - 184,2004.[19] Robert Myers,
Free involutions on lens spaces , Topology, 20(3) : 313 - 318, 1981.[20] Robert Oliver,
A proof of the Conner conjecture , Ann. of Math. (2), 103(3) : 637 - 644, 1976.[21] Pedro L. Q. Pergher, Hemant K. Singh, and Tej B. Singh, On Z and S free actions on spaces of cohomologytype ( a, b ), Houston J. Math., 36(1) : 137 - 146, 2010.[22] A. Pfister and S. Stolz, On the level of projective spaces , Comment. Math. Helv., 62(2) : 286 - 291, 1987.[23] P. M. Rice,
Free actions of Z on S , Duke Math. J., 36 : 749 - 751, 1969.[24] Gerhard X. Ritter, Free Z actions on S , Trans. Amer. Math. Soc., 181 : 195 - 212, 1973.[25] Gerhard X. Ritter, Free actions of cyclic groups of order n on S × S , Proc. Amer. Math. Soc., 46 : 137 - 140,1974.[26] J. H. Rubinstein, Free actions of some finite groups on S . I. Math. Ann., 240(2) : 165 - 175, 1979.[27] Mahender Singh, Orbit spaces of free involutions on the product of two projective spaces , Results Math., 57(1-2) : 53 - 67, 2010.[28] Mahender Singh,
Cohomology algebra of orbit spaces of free involutions on lens spaces , J. Math. Soc. Japan,65(4) : 1055 - 1078, 2013.[29] Mahender Singh,
Free -rank of symmetry of products of Milnor manifolds , Homology Homotopy Appl.,16(1) : 65 - 81, 2014.[30] Yoko Tao, On fixed point free involutions of S × S , Osaka Math. J., 14 : 145 - 152, 1962.[31] Jeffrey L. Tollefson, Involutions on S × S and other -manifolds , Trans. Amer. Math. Soc., 183 : 139 - 152, 1973.[32] A. S. Schwarz, The genus of a fibre space , Trudy Moskov Mat. Obˇsˇc., 11 : 99 - 126, 1962. Translation in Amer.Math. Soc. Trans., 55, 1966, 49 - 140.[33] A. Yu. Volovikov,
On the index of G -spaces , Mat. Sb., 191(9) : 3 - 22, 2000.[34] Chung-Tao Yang, On theorems of Borsuk-Ulam , Kakutani-Yamabe-Yujobˆo and Dyson. II. Ann. of Math. (2),62 : 271 - 283, 1955.
Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER)Mohali, Knowledge City, Sector 81, SAS Nagar, Manauli (PO), Punjab 140306, India.
E-mail address ::