Transfers for ramified coverings in homology * and cohomology Marcelo A. Aguilar & Carlos Prieto1 Instituto de Matem'aticas, UNAM, 04510 M'exico, D.F., Mexico Abstract Making use of a modified version, due to McCord, of the Dold- Thom construction of ordinary homology, we give a simple topological definition of a transfer for ramified covering maps in homology with arbitrary coefficients. The transfer is induced by a suitable map be- tween topological groups. We also define a cohomology transfer which is dual to the homology transfer. This duality allows us to show that our homology transfer coincides with the one given by L. Smith. With our definition of the homology transfer we can give simpler proofs of the properties of the known transfer and of some new ones. 1 Introduction In [16] L. Smith introduced a general class of finite ramified covering maps and constructed for them a transfer in ordinary homology. Later on, in [6] A. Dold gave an alternative construction and characterized ramified covering maps as maps between orbit spaces of the action of a finite group and a subgroup, and giving a modified definition of the transfer. Both definitions are algebraic in nature. These transfers have the property that when composed with the homomorphism induced by the projection of the ramified covering ______________________________ *2000 Math. Subj. Class.: Primary 55R12, 57M12; Secondary 55Q05, 55R35, 57M10 Keywords and phrases: Transfer, ramified covering maps, classifying spaces E-mail addresses: marcelo@math.unam.mx, cprieto@math.unam.mx. 1This author was partially supported by PAPIIT grant No. IN110902. 1 map, they yield multiplication by the multiplicity of the covering in the homology of the base space. There are previous definitions of both the homology and the cohomology transfers for maps between orbit spaces of certain actions of a finite group and a subgroup, see Bredon [4] (and also tom Dieck [5]). These definitions depend on the equivariant structure of the spaces involved. This is analogous to the classical definition of the transfer for standard covering maps of Eckmann [8], which is given at the level of chain complexes, thus it is rather algebraic in nature. In this paper we use a modified version, due to McCord [13], of the Dold- Thom construction of ordinary homology to produce a topological transfer for general ramified covering maps. Namely, we define a transfer that is a map between some topological groups associated to the total and to the base space of the ramified covering map. This codifies in a sense the fact that a transfer can be seen as a multivalued map. We also define a cohomology transfer using models of Eilenberg-Mac Lane spaces that have the structure of topological abelian groups. We apply either transfer to give some results about the homology or cohomology of orbit maps of the action of a group and a subgroup of finite index. In contrast to previous transfers, whose definition, and therefore also their computation, is more complicated than the definition of the homomorphisms induced by the projections of the (ramified) covering maps, our transfer definitions are somehow simpler than the latter, making their computation easy (see (6.1), for instance). In Section 2, for the benefit of the reader, we recall the construction of McCord's topological groups and the nice main results of McCord's paper. These are essential for our definition of the homology transfer and to prove its properties. They also provide models of the Eilenberg-Mac Lane spaces which are (weak) topological abelian groups. This fact will be used later to construct the cohomology transfer. In Section 3 we recall the definition of a ramified covering, and in Section 4 we define the homology transfer and prove some of its properties. In Section 5 we define the cohomology transfer and prove some of its properties. In Section 6 we use the transfers to prove some results about group actions. For instance, we study the homology and the cohomology of the orbit space of an arbitrary action of a compact Lie group in terms of the orbit space of the action of the connected component of the identity element. In Section 7 we show that the homology and the cohomology transfers are dual with respect to the Kronecker product in homology and cohomology. Finally, in Section 8 we prove that our homology transfer coincides with 2 Smith's transfer. 2 McCord's topological groups In this section we recall briefly the spaces B(G, X) introduced by McCord. We find it convenient to use F (X, G) as an alternative notation. Details can be seen in [13] or [1, 6.3.20ff]. In this section we shall work in the category of compactly generated weak Hausdorff spaces (see [13] or [12]) and therein will be all spaces considered. Definition 2.1. Let G be a topological abelian group and let X be a pointed topological space with base point * 2 X. We define F (X, G) as the set of all functions u : X - ! G such that u(*) = 0 and u(x) = 0 for all but a finite number of elements x 2 X. If these elements are x1, . .,.xn and the valuesPof u at each of them are g1, . .,.gn, it is sometimes convenient to write u as ni=1gixi = g1x1 + . .+.gnxn. In particular, for any x 2 X, x 6= 0, one may see gx as the element in F (X, G) whose value at x is g and whose value elsewhere is 0 (g* = 0). Of course, these elements gx generate the group F (X, G). Taking G = R to be a commutative ring with 1 and x 2 X, then x 2 F (X, R) can be interpreted as the function whose value at x is 1 and whose value elsewhere is 0. This furnishes F (X, R) with a canonical inclusion X ,! F (X, R). In this case, the elements x 2 F (X, R) generate F (X, R) freely as an R-module. Remark 2.2. In this paper we shall mainly restrict ourselves to abelian groups G with the discrete topology, however we work in this section with general topological abelian groups. The set F (X, G) has a topology that turns it into a pointed space with base point 0 2 F (X, G) the constant function with value 0, as we see below. It is also an abelian group with the obvious addition. It is in fact a (weak) topological abelian group. Consider the natural filtration F0(X, G) F1(X, G) . . .F (X, G) , where Fn(X, G) consists of those functions u that are nonzero on at most n points in X. If G = R, then one can define F1_2(X, R) = X F1(X, R), as ex- plained above. The topology can then be defined as follows. For each n, take the surjection k(G x X)n -! Fn(X, G) given by mapping (g1, x1, . .,.gn, xn) 3 P n to i=1gixi. Here k(G x X)n is the product of n copies of G x X, furnished with the compactly generated product topology, and Fn(X, G) is given the corresponding quotient topology. Then provide F (X, G) with the weak topol- ogy (of the union). Given a pointed map ' : X - ! Y and a (continuous) homomorphism ff : G -! H, one has a unique pointed map F (', ff) : F (X, G) -! F (Y, H) given by _ n ! X Xn (2.3) F (', ff) gixi = ff(gi)'(xi) . i=1 i=1 In other words, F (', ff)(u) is the function whose valuesPat y 2 Y are 0 unless y = '(x) and u(x) 6= 0; in this case, F (', ff)(u)(y) = '(x)=yff(u(x)). This definition turns F into a covariant bifunctor from the category Top *x Topab of pairs consisting of a pointed topological space and a topological abelian group to the category Topab of topological abelian groups. We shall denote F (', 1G ) simply by '* and F (idX , ff) by ff*. The followi* *ng will be useful properties of the functor F : (a) There is a natural isomorphism of topological abelian groups F (X ^ Y, G) -! F (Y, F (X, G)) P P given by mapping u = gi(xi^yi) to (gixi)yi, where gixi 2 F (X, G) is as described above (see [13, 6.13]). (b) There is a natural H-isomorphism (i.e., a homotopy equivalence that is also a homomorphism) of H-groups F (Y, G) -! F ( Y, G) (if G is well pointed), where means the loop space and the (reduced) suspension given by Y = S1 ^ Y . This H-isomorphism yields a group isomorphism oe : [X, F (Y, G)]* -! [X, F ( Y, G)]* ~=[ X, F ( Y, G)]* , where [-, -]* denotes pointed homotopy classes. We call this the sus- pension isomorphism (see [13, 10.4]). 4 By (b), F (Sq, G) -'! F (Sq+1, G), and since F (S0, G) = G, we obtain the following. Theorem 2.4. ([13, 10.6]) The space F (Sq, G) is an Eilenberg-Mac Lane space of type (G, q) that has the structure of an abelian group. || Note 2.5. This is not the first construction of Eilenberg-Mac Lane spaces that yields topological abelian groups (for instance, Milnor [14] shows that his construction of K(G, q) yields always a weak topological abelian group; he shows that if K(G, q) is a countable CW-complex, then it is a topological abelian group). However, McCord's construction is a very convenient one and is easy to give. Using property (b), we have a long exact sequence for the homotopy groups ßq(F (A, G)), ßq(F (X, G)), and ßq(F (X [ CA, G) for a pair (X, A) of the same homotopy type of a CW-pair . This and the previous theorem prove the following. Corollary 2.6. Let G be a discrete abelian group and let (X, A) be a pair of spaces of the same homotopy type of a CW-pair. Then the homotopy groups Hq(X, A; G) = ßq(F (X [ CA, G)) define an ordinary homology theory with coefficients in G. In particular, the groups eHq(X; G) = ßq(F (X, G)) provide the reduced homology groups. More- over, the groups of pointed homotopy classes Hq(X, A; G) = [X [ CA, F (Sq, G)]* define an ordinary cohomology theory with coefficients in G. In particular, the groups eHq(X; G) = [X, F (Sq, G)]* provide the reduced cohomology groups. (If A ,! X is a cofibration, one may of course replace X [ CA by X=A.) || Note that the groups of unpointed homotopy classes Hq(X; G) = [X, F (Sq, G)] provide the unreduced cohomology groups. Remark 2.7. If we assume that X is paracompact (instead of compactly generated weak Hausdorff of the same homotopy type of a CW-complex), then the groups [X, F (Sq, G)]* yield the ~Cech cohomology groups H~q(X; G) (see [11]). 5 P m P m Lemma 2.8. The map " : F (X, G) -! G given by i=1gixi 7! i=1gi is well defined and continuous. In particular, " : F (S0, G) -! G is a homeo- morphism. Proof: This follows easily from the fact that the restriction "n : Fn(X, G) -! G of " is continuous, since its composite with the identification (X xG)n -! Fn(X, G) is obviously continuous. || Another useful property of the functor F is that one has a well-defined continuous pairing (2.9) F (X, G) x F (Y, H) -! F (X ^ Y, G H) given by _ ! X X X gixi, hjyj 7- ! (gi hj)(xi^ yj) , i j i,j (see [13, 11.6]). If, in particular, G = H = R is a commutative ring with 1, with m : R R -! R as the ring multiplication, then composing (2.9) with m*, we obtain another pairing (2.10) F (X, R) x F (Y, R) -! F (X ^ Y, R) . Using (2.10), one obtains products in homology and cohomology. We shall be interested in the following. Proposition 2.11. One has cap-products Hq(X; R) Hk(X; R) -`! Hk-q(X; R) , if X is 0-connected and q k, and Hq(X; R) Hk(X; R) -`! Hq-k(X; R) , if k q. In particular, if k = q one has a Kronecker product <-,-> (2.12) Hq(X; R) Hq(X; R) - ! R . 6 Proof: Taking smash-products and the pairing (2.9) we have [X+ , F (Sq, R)]* x [Sk, F (X+ , R)]*_//[X+ ^ Sk, F (Sq, R) ^ F (X+ , R)]* X X X X X X | ~ XX X X X X,, fflffl|| [ kX+ , F ( qX+ , R)]* . If q k, using oe-q of property (b), we desuspend q times. Composing ~ with the homomorphism [ k-qX+ , F (X+ , R)]* -! [Sk-q, F (X+ , R)]* induced by the pointed inclusion S0 -! X+ that sends -1 to some point x-1 in the path-connected space X, we obtain the homology `-product `: [X+ , F (Sq, R)]* x [Sk, F (X+ , R)]* -! [Sk-q, F (X+ , R)]* . On the other hand, if k q, using oe-k , we desuspend k times. And then, composing ~ with the homomorphism [X+ , F ( q-kX+ , R)]* -! [X+ , F (Sq-k, R)]* induced by the obvious map X+ -! S0, we obtain the cohomology `- product `: [X+ , F (Sq, R)]* x [Sk, F (X+ , R)]* -! [X+ , F (Sq-k, R)]* . In order to obtain the Kronecker product <-, ->, we take q = k and consider the composite [X+ , F (Sq, R)]*x[Sq, F (X+ , R)]* -`! [X+ , F (S0, R)]* -! [S0, F (S0, R)]* =* * R , where the last arrow is induced by the pointed inclusion S0 -! X+ , and the equality follows from the bijection " : F (S0, R) -! R given in Lemma 2.8. || 3 Ramified coverings We recall L. Smith's definition of a ramified covering map (see [16]). We shall need the concept of nth symmetric product of Y defined by SP nY = Y n= n, where n acts on the product of n copies of Y by permuting the coordinates. We denote its elements by . 7 Definition 3.1. An n-fold ramified covering map is a continuous map p : E - ! X together with a multiplicity function ~ : E - ! N such that the following hold: (i)The fibers p-1(x) are finite (discrete), x 2 X. P (ii)For each x 2 X, e2p-1(x)~(e) = n. (iii)The map 'p : X -! SP nE given by 'p(x) = , ~(e1) ~(em ) where p-1(x) = {e1, . .,.em }, is continuous. Remark 3.2. Given an n-fold ramified covering map p : E - ! X with multiplicity function ~, one can construct an n-fold ramified covering map p+ : E+ -! X+ , where Y + = Y t {*} for any space Y and p+ extends p by defining p+ (*) = * and the multiplicity function ~+ extends ~ by setting ~+ (*) = n. More generally, given a (closed) subspace A X, one can con- struct an n-fold ramified covering map p0: E0 -! X=A, where E0 = E=p-1A, p0 is the map between quotients and the multiplicity function ~0 coincides with ~ off p-1A and is extended by setting ~0(*) = n, if * is the base point onto which p-1A collapses. __ __ Another useful construction is the following. Let E = E t X and __p: E - ! X be such that __p|E = p and __p|X = idX . Then __pis an (n + 1)-fold ramified covering map with the obvious multiplicity function. On the other hand, given a map F : Y -! X, one can construct the induced n-fold ramified covering map F *(p) : F *(E) - ! Y by taking the pullback F *(E) = {(y, e) 2 Y x E | F (y) = p(e)} and F *(p) = projY. The induced multiplicity function F *(~) : F *(E) -! N is given by F *(~)(y, e) = ~(e). Call eF: F *(E) -! E the projection projE. Examples 3.3. Typical examples of ramified covering maps are the follow- ing: 1. Standard covering maps with finitely many leaves. 2. Orbit maps E= 0- ! E= for actions of a finite group on a space E and 0 . They can be considered as [ : 0]-fold ramified covering 8 maps by taking ~(e 0) = [ e : 0e], where e and 0edenote the isotropy subgroups of e 2 E for the action of and the restricted action of 0, and [ : 0] and [ e : 0e] denote the corresponding indexes. In fact, Dold [6] proves that all ramified covering maps are of this form for = n and 0= n-1. 3. Branched covering maps on manifolds, namely open maps p : Md -! Nd, where Md and Nd are orientable closed topological manifolds of dimension d, p has finite fibers and its degree is n. Indeed, Berstein and Edmonds [3] prove that p is of the form E= 0 -! E= , with [ : 0] = n, so that, by 2., p is in fact an n-fold ramified covering map. An interesting special case of this is given by Montesinos [15] and Hilden [10], who show that for any closed orientable 3-manifold M3, there is a branched covering map p : M3 -! S3 of degree 3. 4. It will be of particular interest to consider the following example. Let B be a space and ßB : Bn x n __n-! SP nB, where __n= {1, 2, . .,.n} and x n represents the twisted product, be given by ßB = . Then ßB is an n-fold ramified covering map with mul- tiplicity function ~B : Bn x n __n-! N given by ~B = #{j | bj = bi} (see [16]). 4 The homology transfer We shall define now the homolgy transfer. Our spaces in this section will be compactly generated weak Hausdorff spaces. Definition 4.1. Let p : E - ! X be an n-fold ramified covering map with multiplicity function ~. Define the pretransfer tp : F (X, G) -! F (E, G) by tp(u) = eu, P n where eu(e) = ~(e)u(p(e)). In other words, if u = i=1gixi 2 F (X, G), then X tp(u) = ~(e)gie . p(e)=xi i=1,...,n Remark 4.2. The pretransfer tp : F (X, G) - ! F (E, G) is clearly a ho- momorphism of topological groups and it is thus convenient to see what it 9 does to generators. Namely, if gx is the function in F (X, G) such that it is zero everywhere, with the exception of x, where its value is g, then it is a generator and the pretransfer satisfies ( ~(e)g if p(e) = x, i.e., if e 2 p-1(x) tp(gx)(e) = ~(e)gx(p(e)) = 0 otherwise. Hence, the only points where tp(gx) is nonzero are the elements of p-1(x) = {e1, e2, . .,.er}, that is, tp(gx)(e1) = ~(e1)g , tp(gx)(e2) = ~(e2)g , . .,.tp(gx)(er) = ~(er)g , and thus tp(gx) = ~(e1)ge1 + ~(e2)ge2 + . .+.~(er)ger. We shall prove below that tp is continuous. Hence, on homotopy groups, the map tp induces the homolgy transfer øp : eHq(X; G) -! eHq(E; G) . We have the following. Proposition 4.3. Let p : E -! X be an n-fold ramified covering map with multiplicity function ~ : E -! N, where E and X are pointed spaces. Then the pretransfer tp : F (X, G) -! F (E, G) is continuous. Proof: Since F (X, G) has the topology of the union of the subspaces . . .Fr(X, G) Fr+1(X, G) . . .F (X, G) , tp is continuous if and only if the restriction tp|Fr(X,G)is continuous for each r 2 N. Denote by k(X x Y ) the product of X and Y with the compactly gen- erated topology. Then we have a quotient map qr : k((G x X)r) i Fr(X, G) for each r. Define ffi : G x X - ! Fn(X, G) by ffi(g, x) = tpq1(g, x) = tp(gx), and ff : G x X -! (G x X)n= n by ff(g, x) = [(g, e1), . .,.(g,,e1).,.(.g, em ), . .,.(g,]em,) ________-z_______" ________-z_______" ~(e1) ~(em ) where p-1(x) = {e1, . .,.em }. For each g 2 G, let ig : X - ! G x X be given by ig(x) = (x, g), and let jg : En= n - ! (G x X)n= n be given by 10 jg[e1, . .,.en] = [(g, e1), . .,.(g, en)]. Then ff O ig = jg O 'p, where 'p : X* * -! En= n. Since jg and 'p are continuous and G is discrete, ff is continuous and k(ff) : k(G x X) -! k((G x X)n= n) is also continuous. Since G x X is compactly generated, k(GxX) = GxX. There is a natural homeomorphism k((G x X)n= n) k((G x X)n)= n (indeed, it is straightforward to show that the orbit space of the action of a finite group on a compactly generated weak Hausdorff space is again a compactly generated weak Hausdorff space). Therefore, the map k(ff) : G x X -! k((G x X)n)= n is continuous. The quotient map qn factors through the quotient map q0n: k(G x X)n i k((G x X)n)= n, yielding the following commutative diagram, q0n n k((G x X)n) ____////_k((G x X) )= n lll qn|| llllll fflfflfflffl|jnuuuullll Fn(X, G) , where æn is also a quotient map. Now, ffi makes the following diagram commute, k((G6x6X)n)= n nnn k(ff)nnnnn jn|| nnnnn fflfflfflffl| G x X ___ffi_//_Fn(X, G) , therefore, ffi is continuous. In order to show that tp|Fr(X,G)is continuous, consider the diagram k(ffir) k((G x X)r) _____//k(Fn(E, G) x . .x.Fn(E, G)) qr|| P|ri=1| fflfflfflffl| fflffl| Fr(X, G) ____t___________//F (E, G) , p|Fr(X,G) P r where i=1is the operation in F (E, G), which is a topological abelian group in the compactly generated topology, and hence it is continuous. Since also ffi is continuous, and qr is a quotient map, tp|Fr(X,G)is continuous. || Corollary 4.4. Let p : E - ! X be an n-fold ramified covering map with multiplicity function ~ : E -! N, where E and X are pointed CW-complexes. Then there is a homology transfer øp : eHq(X; G) -! eHq(E; G). || 11 Remark 4.5. Besides the transfer øp defined above, for every integer k there is another homology transfer kø given by (kø)p(,) = k . øp(,), , 2 Hq(X; G). This transfer, in turn, is determined by the pretransfer (kt)p : F (X, G) -! F (E, G) given by (kt)p(u) = k . tp(u), u 2 F (X, G). Example 4.6. For the ramified covering map ßB : Bn x n __n-! SP nB of 3.3, the homology transfer is given as follows. We first compute tiB : F (SP nB, G) -! F (Bn x n __n, G) on the generators. Set b = (b1,_._.,.b1_-z___", b2,_._.,.b2_-z___", . .,.br,_._.,.br_-z_* *__") 2 Bn , i1 i2 ir where i1 + i2 + . .i.r= n. Then ß-1B = {, . .,.} . Therefore, tiB(g) = ~g + ~g + . .+. + ~g = i1g + i2g + . .+.irg = g + + . .<.b,+i1> ____________-z___________" i1 + + + . .+..+. ___________________-z__________________" i2 + + + . .+. ____________-z___________" ir = g + . .+. + + . .<.b, i1 + i2> + . .+. + + = g + g + . .+.g , hence (4.7) tiB(g) = g) + . .+.g) . 12 P k Thus, in general, if fi = i=1gi, then (k,n)X tiB(fi) = gi , (i,l)=(1,1) since by varying l from 1 to n, the fiber elements over , namely , are repeated ~B times. Remark 4.8. Given an n-fold ramified covering map p : E - ! X with multiplicity function ~ : E -! N, and a (closed) subspace A X, we have the restricted ramified covering_map pA : EA - ! A, EA = pA , and the quotient ramified covering map __p: E - ! X=A, as described in Remark 3.2. The following diagram obviously commutes: " __ EA Ø____//_E_____////_E pA || p || |_p| fflffl|Øfflffl|" fflffl| A _____//_X___////_X=A . Thus the diagram above yields F (A, G)______//F (X, G)___//_F (X=A, G) tA|| t|| |_t| fflffl| fflffl| fflffl|_ F (EA , G)_____//F (E, G)____//_F (E , G) , _ where the horizontal arrows are_obvious and tA , t, and tare the corresponding pretransfers. Therefore, using t, we have a relative homology transfer øp : Hn(X, A; G) -! Hn(E, EA ; G), and by the commutativity of the diagram, also this transfer maps the long exact sequences of (X, A) into the long exact sequence of (E, EA ), provided that the inclusion A ,! X is a closed cofibration (in general it is also true by constructing an adequate ramified covering over X [ CA). The following theorems establish the fundamental properties of the trans- fer. Theorem 4.9. The composite p* O øp : eHn(X; G) -! eHn(X; G) is multiplication by n. 13 The proof follows immediately from the following proposition. Proposition 4.10. If p : E -! X is an n-fold ramified covering map, then the composite tp p* F (X, G) -! F (E, G) -! F (X, G) is multiplication by n. P n iP n j Proof: If u = i=1gixi 2 F (X, G), then p*tp(u) = p*tp i=1gixi = P P n P P n p(e)=xi, i=1,...,n~(e)gixi = i=1gixi p(e)=xi~(e) = n i=1gixi = n . u. * * || The invariance under pullbacks is given by the following. Theorem 4.11. Assume that F : X -! Y is continuous and that g : G -! H is a homomorphism of discrete abelian groups. Then the following diagram commutes: fiF*(p) Hq(Y ; G)_____//Hq(F *(E); G) F*|| |eF*| |fflffl fflffl| Hq(X; G) __fip//_Hq(E; G) , where F *(E) -! Y is the n-fold ramified covering map induced by p : E -! X over F . As for the previous theorem, the proof follows immediately from the next proposition. Proposition 4.12. If p : E - ! X is an n-fold ramified covering map and F : X -! Y is continuous, then the following diagram commutes. tF*(p) F (Y, G)_____//F (F *(E), G) F*|| Fe*|| fflffl| fflffl| F (X, G) __tp__//_F (E, G) . 14 P n Proof: Let v = i=1giyi 2 F (Y, G). Then tF*(p)(v) 2 F (F *(E), G) is such that 0 1 B X C eF*tF*(p)(v) = eF*B F (~)(y, e)g (y, e)C @ * i A F*(p)(y,e)=yi i=1,...,n X = ~(e)gieF(y, e) F*(p)(y,e)=yi i=1,...,n X = ~(e)gie p(e)=F(yi) i=1,...n = tp (F*(v)). || One further property of the homology transfer that is useful is the fol- lowing. Proposition 4.13. Let f : B - ! C be continuous and consider the com- mutative diagram __ fnx n1_n n __ (4.14) Bn x n n __________//_C x n n iB || |iC| fflffl| fflffl| SP nB _____SPnf____//SPnC . Then the following diagram commutes: (fnx n1_n)* n __ F (Bn x nO__n,OG)_________//_F (C OxOn n, G) tßB || |tßC| | | F (SP nB, G)____(SPnf)___//F (SP nC, G) . * The proof is fairly routinary and follows easily using the description of the transfers given in Example 4.6. || In 4.10 we computed the composite p*O tp. The opposite composite tpO p* is also interesting. An immediate computation yields the following. 15 Proposition 4.15. Let p : E -! X by an n-fold ramified covering map with multiplicity function ~. Then the composite p* tp F (E, G) -! F (X, G) -! F (E, G) is given by X tpp*(v)(e) = ~(e0)v(e0) , p(e0)=p(e) for any v 2 F (E, G). || In the case of an action of a finite group on E and X = E= , we have the following consequence. P Corollary 4.16. For v 2 F (E, G) one has tpp*(v)(e) = fl2 v(fle). There- fore, the composite p* tp F (E= , G) -! F (E, G) -! F (E= , G) P is given by tpp*(v) = fl2 fl*(v). Proof.Just observe that the element fle is repeated in the sum ~(e) = | e| __ times. |__| The two previous results yield the following in homology. Theorem 4.17. Let p : E - ! X by an n-fold ramified covering map with multiplicity function ~. Then the composite p* fip Hq(E; G) -! Hq(X; G) -! Hq(E; G) is given by øpp*(y) = y0, where y0= [v0] 2 ßq(F (E, G)), and X v0(s)(e) = ~(e0)v(s)(e0) p(e0)=p(e) where y = [v] 2 ßq(F (E, G)) and s 2 Sq. || Corollary 4.18. For an action of a finite group on E and X = E= one has that the composite p* fip Hq(E; G) -! Hq(E= ; G) -! Hq(E; G) P is given by øpp*(y) = fl2 fl*(y). 16 Remark 4.19. Considering an action of H on E and a subgroup K H, one has different ramified covering maps as depicted in E F q0xxxx FFFqF xx FFF --xxx ""F E= 0 _______0____//E= . q One may easily compute several combinations of the maps induced by these covering maps and their transfers. Another interesting property of the transfer is the relationship given by computing the transfer of the composition of two ramified covering maps. Before giving it we need the following. Definition 4.20. Let p : Y -! X be an n-fold ramified covering map, with multiplicity function ~ : Y -! N and let q : Z -! Y be an m- fold ramified covering map, with multiplicity function : Z - ! N. Then the composite p O q : Z - ! X is an mn-fold ramified covering map, with multiplicity function , : Z -! N given by ,(z) = (z)~(q(z)). In order to verify that this composite is indeed an mn-fold ramified covering map, consider the wreath product n s m , defined as the semidirect product of n and ( m )n, where n acts on ( m )n by permuting the n factors. We have an action (Zm x . .x.Zm ) x n s m -! Zm x . .x.Zm given by (i1, . .,.in) . (oe, ø1, . .,.øn) = (iff(1). ø1, . .,.iff(n). øn), where ii 2 Z* *m . Then we have the following diagram, where all maps are open qx...xq m m Zm x . .x.Zm __________//_Z = m x . .x.Z = m i || i0|| fflffl| fflffl| (Zm )n= n s m ` ` ` ` ` ` ` `//SPn(SP mZ) . One may easily show that ß is compatible with ß0O (q x . .x.q). Therefore, there is a homeomorphism Xmn = n s m SP n(SP mZ) and hence one has a canonical quotient map æ : SP n(SP mZ) -! SP mnZ. Then one can easily j mn verify that 'pOq= æ O SP n('q) O 'p : X - ! SP n(SP mZ) -! SP Z. Thus 'pOqis continuous. The homology transfer behaves well with respect to composite ramified covering maps. 17 Theorem 4.21. The following hold: tp tq tpOq= tq O tp : F (X; G) -! F (Y ; G) -! F (Z; G) ; fip fiq øpOq= øq O øp : Hk(X; G) -! Hk(Y ; G) -! Hk(Z; G) . Proof: As before, the second formula follows from the first. So, if u 2 F (X; G), v 2 F (Y ; G), w 2 F (Z; G), then v = tp(u) if v(y) = ~(y)u(p(y)), and w = tq(v) if w(z) = (z)v(q(z)). Hence (tqtp(u))(z) = tq( (z)v(q(z)) = (z)~(q(z))u(pq(z)) = ,(z)u((p O q)(z)) = tpOq(u)(z). || Corollary 4.22. Given an n-fold ramified covering map p : E - ! X with multiplicity function ~ and an integer l, there is an ln-fold ramified covering map pl: E -! X such that pl= p and ~l(e) = l~(e), e 2 E. Then tpl= ltp : F (X; G) -! F (E; G) and øpl= løp : Hk(X; G) -! Hk(E; G). Proof: Consider the l-fold ramified covering map q : E - ! E such that q = idE and (e) = l for all e 2 E. Hence pl = p O q. Then apply Theorem 4.21. || Remark 4.23. The ln-fold covering map plobtained from p is a sort of spuri- ous ramified covering map, since the multiplicity of p is artificially multipli* *ed by l. It is interesting to remark that the previous result shows that the trans- fer of this new ramified covering map pl is just the corresponding multiple of the transfer of the original ramified covering map p. Thus on this sort of artificial ramified covering maps, the transfer remains essentially unchanged. 5 The cohomology transfer In this section we define the chomology transfer and prove some of its prop- erties. Definition 5.1. Let p : E - ! X be an n-fold ramified covering map with multiplicity function ~, where E and X are compactly generated weak Haus- dorff spaces of the same homotopy type of CW-complexes. Define its coho- mology transfer øp : Hq(E; G) = [E, F (Sq, G)] -! [X, F (Sq, G)] = Hq(X; G) 18 P by øp([eff]) = [ff], where ff(x) = p(e)=x~(e)eff(e), x 2 X. To see that the m* *ap ff is continuous and that its homotopy class depends only on the homotopy class of eff, observe that ff is given by the composite 'p n SPneff n q q ff : X -! SP E - ! SP F (S , G) -! F (S , G) , where the last map is given by the group structure on F (Sq, G). Using the fact that X has the homotopy type of a CW-complex, similar arguments to those used in the proof of 4.3 show that ff is continuous. We might write øpninstead of øp when we wish to remark the multiplicity n of the ramified covering map p. Remark 5.2. We might assume that E and X are paracompact spaces in- stead of compactly generated weak Hausdorff spaces of the same homotopy type of a CW-complex. In this case, the same definition yields a transfer that is a homomorphism between ~Cech cohomology groups øp : ~Hq(E; G) -! ~Hq(X; G) , (see Remark 2.7), provided that G is an at most countable coefficient group. Note 5.3. In order to define the cohomology transfer, the only property of the Eilenberg-Mac Lane spaces given by F (Sq, G) required, is the fact that they are (weak) topological abelian groups. Similarly to the homology transfer, the cohomology transfer has the fol- lowing fundamental properties. Theorem 5.4. The composite øpnO p* : Hk(X; G) -! Hk(X; G) is multiplication by n. Proof: If [ff] 2 [X, F (Sk, G)], then øpp*(ff) = øp(ff O p) : X - ! F (Sk, G), P iP j and øp(ffOp)(x) = p(e)=x~(e)ffp(e) = p(e)=x~(e) ff(x) = n.ff(x). Thus øpp*([ff]) = n . [ff]. || 19 Theorem 5.5. Let p : E - ! X be an n-fold ramified covering map and assume that F : Y -! X is continuous. Then the following diagram com- mutes: p Hq(E; G) ___fi__//_Hq(X; G) eF*|| |F*| fflffl| fflffl| Hq(F *(E); G) ___*_//Hq(Y ; G) , fiF (p) where F *(p) : F *(E) -! Y is the n-fold ramified covering map induced by p : E -! X over F . Proof: Let eff: E -! F (Sq, G) represent an element in Hq(E; G). Then the map X X y 7- ! F *(~)(y, e)eff(y, e) = ~(e)eff(y, e) F*(p)(y,e)=y p(e)=F(y) *(p) * *p q that represents øF eF(eff), clearly represents also F ø ([eff]) 2 H (Y ; G). || In 5.4 we computed the composite øp O p*. The opposite composite p*O øp is also interesting. As it was the case for the homology transfer, an immediate computation yields the following results for the cohomology transfer. Proposition 5.6. Let p : E -! X be an n-fold ramified covering map with multiplicity function ~. Then the composite p q p* q Hq(E; G) -fi!H (X; G) -! H (E; G) is given as follows. Take ['] 2 Hq(E; G) = [E, F (Sq, G)], then p*øp['] is represented by the map '0: E -! F (Sq, G) given by X '0(e) = ~(e0)e0. p(e0)=p(e) || In the case of an action of a finite group on E and X = E= , we have the following consequence. Corollary 5.7. If , 2 Hq(E; G), then X p*øp(,) = fl*(,) 2 Hq(E; G), . fl2 20 Proof: Just observe that in the sum the element fl*(,) is repeated ~(e) = | e| times. || Generalizations and further properties of the cohomology transfer are studied in [2]. 6 Some applications of the transfers First we start considering a standard n-fold covering map p : E - ! X. In this case, the pretransfer (and thus also the transfer in homology) has a particularly nice definition. Since the multiplicity function ~ : E - ! N is constant ~(e) = 1, the transfer tp : F (X, G) -! F (E, G) is given by (6.1) tp(u)(e) = u(p(e)) . This fact has a nice consequence. Theorem 6.2. Let be a finite group acting freely on a Hausdorff space E. Then the orbit map p : E - ! E= is a standard covering map, and its pretransfer induces an isomorphism ~= tp : F (E= , G) -! F (E, G) , where the second term represents the fixed points under the induced -action on F (E, G). Consequently, the pretransfer yields an isomorphism ~= Hq(E= ; G) -! ßq(F (E, G) ) , for all q. Proof.We assume that the projection p : E -! E= maps the base point to the base point. The pretransfer tp is a monomorphism. Namely, if tp(u) = 0, then, by (6.1), u(p(e)) = tp(u)(e) = 0 for all e 2 E. Since p is surjective, u = 0. On the other hand, obviously tp(u) 2 F (E, G) for all u 2 F (E= , G). To see that it is an epimorphism, take any v 2 F (E, G) . Then v(e) = v(efl) for all fl 2 , and thus v determines a well-defined element u 2 F (E= , G) __ by u(e ) = v(e). Then clearly tp(u) = v. |__| 21 In what follows, we use the fundamental properties 4.9 and 4.18, and 5.4 and 5.7 of both the homology and the cohomology transfers to prove some results about the homology and cohomology of orbit maps between orbit spaces of the action of a topological group and a subgroup 0 of finite index on a compactly generated weak Hausdorff space of the same homotopy type of a CW-complex (and a corresponding result in ~Cech cohomology for a paracompact space). Before starting we need to recall Dold's definition of an n-fold ramified covering map [6]. It is a finite-to-one map p : E - ! X together with a continuous map _p : X -! SP nE such that (i)for every e 2 E, e appears in the n-tuple _p(p(e)) = , and (ii)SP n(p)_p(x) = 2 SP nX. This definition is equivalent to Smith's (see 3.1), by setting 'p = _p and defining ~(e) as the number of times that e is repeated in _p(p(e)). We have the following interesting result. Proposition 6.3. Let be a topological group acting on a space Y on the right and let 0 be a subgroup of finite index n. Then the orbit map p : Y= 0- ! Y= is an n-fold ramified covering map. Proof: There is a commutative diagram Y x __________//Y | | | | fflffl| fflffl| Y x ( = 0) ____//_Y= 0, where the top map is the action and the vertical maps are the quotient maps. Take the adjoint map of , j : Y - ! Map ( = 0, Y= 0). The function space Map ( = 0, Y= 0) has a right -action given as follows. For f : = 0- ! Y= , take (f . fl)[fl1] = f(fl[fl1]) = f[flfl1]. The map j is then -equivariant and thus induces a map __j: Y= -! Map ( = 0, Y= 0)= . On the other hand, if we identify = 0 with the set n_= {1, . .,.n}, then we have a homeomorphism Map ( = 0, Y= 0)= Map (n_, Y= 0)= n = SP n(Y= 0) . 22 Let _p : Y= -! SP n(Y= 0) be __jfollowed by the previous homeomorphism. Then _p satisfies conditions (i) and (ii) and thus p is an n-fold ramified covering map. || We apply the results 4.10 and 4.16 that we have for the pretransfer to the n-fold ramified covering described above to obtain the following. Proposition 6.4. Let Y be a space with an action of a topological group and let 0 be a subgroup of finite index n. Assume that R is a ring where the integer n is invertible. Then p* : F (Y= 0, R) -! F (Y= , R) is a split (continuous) epimorphism. Moreover, if is finite and its order m is invertible in R, then the kernel of p* is the complement in F (Y= 0, R) of the invariant subgroup F (Y= 0, R) under the induced action of . Thus in this case F (Y= , R) ~=F (Y= 0, R) ; in particular, if is finite and 0 is trivial, then m = n and F (Y= , R) ~=F (Y, R) . Proof: By 4.10 applied to the n-fold ramified covering p : Y= 0 -! Y= , p* O tp : F (Y= 0, R) - ! F (Y= 0, R) is multiplication by n, hence it is an isomorphism, and consequently p* is a split epimorphism. Moreover, if is finite of order m, by 4.16, we have that tpO p* : F (Y= 0, R) -! F (Y= 0, R) is multiplication by m. So, if m is invertible in R, then p* : F (Y= 0, R) -! F (Y= , R) is an isomorphism. || As an immediate consequence of the result above, or applying 4.9 and 4.18, we obtain the following two well-known results (cf. [16, 2.5], [4], [5]). Theorem 6.5. Let Y be a space with an action of a topological group and let 0 be a subgroup of finite index n. Assume that R is a ring where the integer n is invertible. Then p* : Hq(Y= 0; R) -! Hq(Y= ; R) is a split epimorphism. Moreover, if is finite and its order m is invertible in R, then the kernel of p* is the complement of Hq(Y= 0; R) in Hq(Y= 0; R). Thus in this case Hq(Y= ; R) ~=Hq(Y= 0; R) ; and in particular, Hq(Y= ; R) ~=Hq(Y ; R) . || 23 Similarly, 5.4 and 5.7 one has for cohomology the following. Theorem 6.6. Let Y be a space with an action of a topological group and let 0 be a subgroup of finite index n. Assume that R is a ring where the integer n is invertible. Then p* : Hq(Y= ; R) -! Hq(Y= 0; R) is a split monomorphism. Moreover, if is finite and its order m is invertible in R, then the image of p* is Hq(X; R) . Thus in this case Hq(Y= ; R) ~=Hq(Y= 0; R) ; and in particular, Hq(Y= ; R) ~=Hq(Y ; R) . || Remark 6.7. One may take a paracompact space Y with an action of a topological group and obtain for ~Cech cohomology an analogous result, namely p* : ~Hq(Y= ; R) -! ~Hq(Y= 0; R) is a split monomorphism, and ~Hq(Y= ; R) ~=H~q(Y= 0; R) . A nice application of the previous ideas is the following generalization of a well-known result of Grothendieck [9] (in the case Y = E ). Theorem 6.8. Let be a compact Lie group and let 1 be the component of 1 2 . Let R be a ring where n = [ , 1] is an invertible element. For an action of on a topological space Y , one has Hq(Y= ; R) ~=Hq(Y= 1; R) = 1, Hq(Y= ; R) ~=Hq(Y= 1; R) = 1, ~Hq(Y= ; R) ~=H~q(Y= 1; R) = 1, the last two according to what kind of a space Y is. || 7 Duality between the homology and cohomology transfers In this section we compare the homology transfer with the cohomology trans- fer. 24 Given an n-fold ramified cover p : E - ! X with multiplicity function ~ : E - ! N, we can extend it to the n-fold ramified covering map p+ : E+ - ! X+ as explained in Remark 3.2. Consider the cohomology transfer øp : Hq(E; G) = eHq(E+ ; G) -! eHq(X+ ; G) = Hq(X; G) , and consider also the homology transfer øp : Hq(X; G) = eHq(X+ ; G) -! eHq(E+ ; G) = Hq(E; G) as given in Definition 4.1. Theorem 7.1. Let p : E -! X be an n-fold ramified covering map with mul- tiplicity function ~ : E -! N and E path connected, and let øp : Hq(X; R) -! Hq(E; R) and øp : Hq(E; R) -! Hq(X; R) be its homology and cohomolgy transfers. If , 2 Hq(X; G) and e,2 Hq(E; G), then <øp(,), e,>E = <,, øp(e,)>X 2 R , for the Kronecker products for E and X, respectively, and R a commutative ring with 1 (see (2.12)). Proof: We have to prove the commutativity of the following diagram: [X+ , F (Sq, R)]*OxO[Sq, F (X+ , R)]*`//_[X+ , FM(S0, R)]* | MMM fipx1| MMM | MMM + q q + M&&M [E , F (S , R)]* x [S , F (X , R)]* q8R8q qqq 1xfip|| qqqq fflffl| qqq [E+ , F (Sq, R)]* x [Sq, F (E+ ,_R)]*`//_[E+ , F (S0, R)]* By the naturality of the construction of the pretransfers and the definition of the `-product (see Proposition 2.11), it is fairly easy to check that this commutativity follows from the commutativity of the following: [X+ , F (X+O,OR)]*___//_[S0, F (X+M, R)]* | MMM fip| MMM | MMM + + M&&M [E , F (X , R)]* q8R8q qqq fip|| qqqq fflffl| qqq [E+ , F (E+ , R)]*___//[S0, F (E+ , R)]* 25 P m(e) Let ffi : E+ -! F (X+ , R) be given by ffi(e) = i=1 ri(e)xi(e), e 2 E. Chasing this element ffi along the top of the diagram, one easily verifies that it maps to the element X m(e)X d = ~(e) ri(e) , p(e)=x-1 i=1 while chasing it along the bottom of the diagram, it maps to the element m(e-1)X X m(e-1)X d0= ri(e-1) ~(ei) = n ri(e-1) . i=1 p(ei)=xi(e-1) i=1 P m(e) Call æ(e) = i=1 ri(e). Since æ = Ö ffi, by 2.8 this defines a continuous map æ : E -! R, but since E is path connected and R is discrete, æ is constant with value rffi2 R. Hence X d = ~(e)æ(e) = n . rffiand d0= næ(e-1) = n . rffi. p(e)=x-1 Thus d = d0 and the diagram commutes. || For simplicity, in what follows we omit the coefficient ring R in homology and cohomology. For the Kronecker product <-, ->Y : Hq(Y ) Hq(Y ) -! R there are induced homomorphisms Y : Hq(Y ) -! Hom (Hq(Y ), R) and Y : Hq(Y ) - ! Hom (Hq(Y ), R) for every space Y , given by (y)(j) = Y and (j)(y) = Y , y 2 Hq(Y ), j 2 Hq(Y ). Corollary 7.2. The following diagrams commute E X q Hq(E) ______//Hom(Hq(E), R) Hq(X) _____//_Hom(H (X), R) fip|| Hom|(fip,1)| fip|| |Hom(fip,1)| fflffl| fflffl| fflffl| fflffl| Hq(X) __X__//Hom(Hq(X), R) , Hq(E) __E_//_Hom(Hq(E), R) , the one on the right-hand side only if øp : Hq(E) -! Hq(X) is a homomor- phism (which is rather seldom the case). || Remark 7.3. Under suitable conditions or are isomorphisms, in whose case one of the transfers determines the other. 26 8 Comparison with Smith's transfer In this section we show that the transfer defined in [16] coincides with ours if we take Z-coefficients. To that end, we first recall his definition of the transfer. It makes use of a result of Moore, that we state below. Recall that Q 1 the weak product e n=1Xn of a family of pointed spaces is the colimit over n of the directed system of spaces X1 ,! X1 x X2 ,! X1 x X2 x X3 ,! . .,. where the inclusions are given by letting the last coordinate be the base point. Moore's result, as it appears in [18], is as follows. Theorem 8.1. (Moore) A connected space X is weakly homotopy equivalent Q to the weak product en 1K(ßn(X), n) of Eilenberg-Mac Lane spaces if and only if the Hurewicz homomorphism hn : ßn(X) - ! Hen(X; Z) is a split monomorphism for all n 1. || Suppose that æn : eHn(X) = eHn(X; Z) -! ßn(X) is a left inverse of hn. The Kronecker product defined in Section 2 determines an epimorphism Hen(X; ßn(X)) -! Hom (Hen(X), ßn(X)) . Let [,n] 2 eHn(X; ßn(X)) = [X, K(ßn(X), n)]* be some preimage of æn. Then the family of pointed maps (,n) defines the weak homotopy equivalence of the previous theorem. Corollary 8.2. If X is a connected topological abelian monoid of the same homotopy type of a CW-complex, then there is a homotopy equivalence X -! eQ n 1K(ßn(X), n). Proof: Since X is a topological abelian monoid, there is a retraction r : SP1 X -! X given by the retractions rn : SP nX -! X , rn = x1 + x2 + . .+.xn . Recall, on the other hand, that by the Dold-Thom theorem one has an iso- morphism ßn(SP 1 X) ~=Hen(X), so that the inclusion i : X ,! SP 1X defines the Hurewicz homomorphism (see [1]). Since r O i = idX, the homomorphism æn = r* : Hen(X) = ßn(SP 1 X) - ! ßn(X) provides a left inverse of the Hurewicz homomorphism hn. Hence, by Moore's theorem, we obtain the re- sult. || 27 Remark 8.3. Note that in the proof above, it is enough to assume that X is a weak topological abelian monoid, i.e., that the product in X is continuous on compact sets. For any space E, the space SP 1E is a weak topological abelian monoid. Thus we have the following. Corollary 8.4. For a connected space E of the same homotopy type of a CW-complex, there is a natural homotopy equivalence wE : SP 1 E - ! Q 1 K(He*(E)) = e n=1K(Hen(E), n). || The definition of Smith's transfer is as follows. Given an n-fold ramified cover p : E - ! X with multiplicity function ~ : E - ! N, consider the following composite: 'p n 1 ' bp: X -! SP E -! SP E -! K(He*(E)) . This map defines a family of elements [bp] 2 He*(X; eH*(E)). On the other hand, the Kronecker product determines a homomorphism _ : eH*(X; eH*(E)) -! Hom (He*(X), eH*(E)) . Smith's transfer is the image p] : eH*(X) -! eH*(E) of [bp] under the homo- morphism _. Theorem 8.5. Let p : E -! X be an n-fold ramified cover with multiplicity function ~ : E - ! N. Then p] = øp : He*(X; Z) -! He*(E; Z), where øp is the transfer in reduced homology. Proof: Consider the following commutative diagram. ~= [E, SPnE]* _____//[E, SP1 E]*___//_eH*(E, eH*(E))___//Hom(He*(E), eH*(E)) fip|| fip|| fip|| Hom|(fip,1)| fflffl| fflffl|~ fflffl| fflffl| [X, SPnE]* _____//[X, SP1 E]*=__//_eH*(X, eH*(E))__//_Hom(He*(X), eH*(E)) The two squares on the left-hand side, where øp represents the cohomology transfer, commute obviously. The one on the right-hand side commutes by Corollary 7.2. Take [i] 2 [E, SPnE]*, where i : E ,! 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