A classification of cohomology transfers for * ramified coverings Marcelo A. Aguilar & Carlos Prieto1 Instituto de Matem'aticas, UNAM, 04510 M'exico, D.F., Mexico Abstract We construct a cohomolgy transfer for n-fold ramified covering maps. Then, we define a very general concept of transfer for ramified covering maps and prove a classification theorem for these transfers. This generalizes Roush's classification of transfers for n-fold ordinary covering maps. We characterize those representable cofunctors which admit a family of transfers for ramified covering maps that have two naturality properties, as well as normalization and stability. This is analogous to Roush's characterization theorem for the case of ordi- nary covering maps. Finally, we classify these families of transfers and construct some examples. In particular, we extend the determinant function in GL (k, C) to a transfer. 1 Introduction In [3], we defined a transfer for ramified covering maps in ordinary cohomol- ogy. We start this paper by giving a transfer homomorphism tp : h(E) = [E, H] - ! h(X) = [X, H] for any topological abelian monoid H and any ramified covering map p : E - ! X. In particular, if H is an Eilenberg- Mac Lane space (modelled by a topological abelian group), then we have ______________________________ *2000 Math. Subj. Class.: Primary 55R12, 57M12; Secondary 55Q05, 55R35, 57M10 Keywords and phrases: Transfer, covering maps, ramified covering maps, classify* *ing spaces E-mail addresses: marcelo@math.unam.mx, cprieto@math.unam.mx. 1This author was partially supported by PAPIIT grant No. IN110902. 1 the cohomology transfer. This transfer is an example of what we shall call (h, k)-transfers, where h and k are representable functors from the homotopy category of spaces to the category of sets, represented by spaces H and K (not necessarily topological abelian groups or H-spaces). We use the properties of the transfer in ordinary cohomology to define the concept of a general (h, k)- transfer for ramified covering maps. We give a classification of these transfers that extends the classification of transfers for ordinary covering maps given by Roush [8]. In particular, the set of (h, k)-transfers has a canonical group structure, when k is group-valued. Our results are applied to the study of transfer families and their classification and to conclude that there are (h, h* *)- transfers if and only if H is a weak product of Eilenberg-Mac Lane spaces. This is particularly interesting, since if one finds an (h, h)-transfer family, then it follows that H has to be a weak product of Eilenberg-Mac Lane spaces. The structure of the paper is as follows. In Section 2 we recall the defi- nition of a ramified covering map given by Smith [9] and define our (h, h)- transfer for h(-) = [-, H]. In Section 3 we give the definition of a general (h, k)-transfer and study its properties. We prove that there is a one-to-one correspondence between (h, k)-transfers for n-fold ramified covering maps and elements in k(SP nH). We prove further that there are nontrivial trans- fers for n-fold ramified covering maps in singular cohomology (for large n) only when the dimensions of the cohomology groups are the same, and that these transfers are classified by the integers. In Section 4 we compare our transfers with transfers for ordinary covering maps and prove that our clas- sification extends Roush's classification. In Section 5 we consider families of (h, k)-transfers for n-fold ramified covering maps for all n and give their classification. Namely, we prove that there is a one-to-one correspondence be- tween families of (h, k)-transfers for ramified covering maps and elements in limnk(SP nH). Analogously to Roush's characterization theorem for the case of ordinary covering maps, we give a characterization of those representable functors which admit a family of transfers. We also show that for singular co- homology, all transfers are determined by the transfers for 2-fold ramified cov- ering maps. We finish the section by giving examples of transfers for functors that are not cohomology theories. In particular, we extend the determinant function det : GL (k, C) -! C*, which yields an element in H2(BGL (k, C)), to a transfer for ramified covering maps ø : VectCk(-) -! VectC1(-). Finally, in Section 6, we study transfers for h(-) = k(-) = H1(-; Z) and prove that the transfers for ordinary covering maps are the same as those for ramified 2 covering maps, i.e., that in this case, one can extend in a unique way the transfers for ordinary covering maps to transfers for ramified covering maps. We conclude that for each n, the group of transfers for n-fold ramified cover- ing maps in 1-cohomology is isomorphic to the group of transfers for ordinary covering maps, and both are isomorphic to Z. 2 Transfers for n-fold ramified covering maps We start by recalling L. Smith's definition of a ramified covering map (see [9]). We shall need the concept of nth symmetric power of Y defined by SP nY = Y_x_._.x.Y-z____"= n , n where n represents the nth symmetric group acting on the product Y x. .x. Yn by permuting coordinates. It is sometimesoconvenient to view SP nY as P k P k i=1miyi 2 Fn(Y, Z) | mi 0, i=1mi n , where Fn(Y, Z) denotes the McCord classifying space (see [3]Pfor a thorough discussion on this). Denote the elements of SP nY either by ki=1miyi, or by , where t* *here are possible repetitions, for instance, y01= . . .= y0m1= y1, y0m1+1= . . .= P k y0m1+m2 = y2, . .,.y0m1+...+mk-1+1= . . .= y0n0= yk, n0 = i=1mi, although the order is irrelevant, since these elements are really nonempty sets with at most n members. Definition 2.1. 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. P (iii)The map 'p : X -! SP nE given by 'p(x) = e2p-1(x)~(e)e is contin- uous. Remark 2.2. 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 3 ~+ (*) = 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) -! Z is given by F *(~)(y, e) = ~(e). Call eF: F *(E) -! E the projection projE. Examples 2.3. Typical examples of these ramified covering maps are orbit maps E -! E=G of actions of a finite group G on a space E. They can be considered as |G|-fold ramified covering maps by taking ~(e) = |Ge|, where Ge denotes the isotropy subgroup of e 2 E and |H| denotes the order of a group H. 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 rep- resents the twisted product, be given by ßB = . Then ßB is an n-fold ramified covering map with multiplicity function ~B : Bn x n __n-! Z given by ~B = #{j | bj = bi} (see [9]). Definition 2.4. Let p : E - ! X be an n-fold ramified covering map with multiplicity function ~. If H is a topological abelian group, define tp : [E, H] -! [X, H] by tp([eff]) = [ff] , P where ff(x) = p(e)=x~(e)eff(e), x 2 X. Let X be a pointed space and G be an abelian (topological) group. Denote by F (X, G) the McCord topological group of functions u : X - ! G such that u(*) = 0 and u(x) = 0 for all but finitely many elements in X. This has the structure of a topological group (see [7] or [3]). If H is an Eilenberg- Mac Lane space of type K(G, q), for instance given by F (Sq, G), then tp is the cohomolgy transfer øp : eHq(E; G) -! eHq(X; G) . 4 Here eHq stands for ordinary cohomology when the spaces involved have the homotopy type of CW-complexes, or for ~Cech cohomology if they are para- compact Hausdorff, provided that either G is countable or the spaces are compactly generated (see [5]). Example 2.5. For the ramified covering map ßB : Bn x n __n-! SP nB of 2.3, the cohomology transfer is as follows. If H is a topological abelian group, then we have similarly that tiB : [Bn x n __n, H] -! [SP nB, H] is given by tp([eff]) = [ff], where Xn ff = fef . l=1 Remark 2.6. Given an n-fold ramified covering map p : E - ! X with multiplicity function ~ : E - ! Z, and a subspace A X, we have the restricted ramified covering map pA : EA - ! A, EA = p-1A, and the quotient ramified covering map p0: E0 -! X=A, as given in Remark 2.2. The following diagram clearly commutes. " EA Ø____//_E_____////_E0 pA || p || |p0| fflffl|Øfflffl|" fflffl| A _____//_X___////_X=A . Then one has cohomology transfers for each covering map. In particular, if A is a subcomplex of a CW-complex X, one has a relative cohomology transfer tp : Hn(E, EA ; G) -! Hn(X, A; G) that fits together into a transformation of the long cohomology exact sequences of the pairs. The following propositions establish the fundamental properties of the transfer. Proposition 2.7. If p : E -! X is an n-fold ramified covering map, then the composite p* tp [X, F] -! [E, F] -! [X, F] is multiplication by n. 5 Proof: If [ff] 2 [X, F], then tpp*(ff) = tp(ffOp) : X -! H, and tp(ffOp)(x) = P iP j p * p(e)=x~(e)ffp(e) = p(e)=x~(e) ff(x) = n . ff(x). Thus t p ([ff]) = n . [f* *f]. || As a consequence, we obtain Theorem 5.4 in [3]. We also obtain the following. Proposition 2.8. Let Zn act on Xn by cyclic permutation of coordinates, and take the quotient map p : Xn -! Xn=Zn. If the prime q does not divide n, then p* : Hl(Xn=Zn; Zq) -! Hl(Xn; Zq) is a split monomorphism. Proof: The map p : Xn -! Xn=Zn is an n-fold ramified covering map. Take its transfer given by the additive structure of K(Zq, l). Then øp O p* is multiplication by n, thus an isomorphism. Hence p* is a split monomorphism. || The invariance under pullbacks is given by the following. Proposition 2.9. If p : E -! X is an n-fold ramified covering map, F is a topological abelian group, and F : X -! Y is continuous, then the following diagram commutes. p [E, F]___t___//[X, F] Fe*|| F*|| fflffl| fflffl| [F *(E), F]__*_//[Y, F] . tF (p) Proof: Let eff: E -! F represent an element in [E, F]. 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 that represents tF Fe ([eff]), clearly represents also F t ([eff]) 2 [Y, F]. * * || As a consequence, we obtain Theorem 5.5 in [3]. One further property of the cohomology transfer that will be useful below is the following. 6 Proposition 2.10. Let f : B - ! C be continuous and consider the com- mutative diagram __ fnx n1_n n __ (2.11) Bn x n n __________//_C x n n iB || |iC| fflffl| fflffl| SP nB _____SPnf____//SPnC . Then the following diagram commutes: ßC n [Cn x n __n, F]t___//_[SP C, F] (fnx n1_n)*|| |(SPnf)*| fflffl|_ fflffl| [Bn x n n, F] _tßB//_[SP nB, F] . The proof is fairly routinary and follows easily using the description of the transfer given in Example 2.5. || In 2.7 we computed the composite tp O p*. The opposite composite p* O tp is also interesting. An immediate computation yields the following. Proposition 2.12. (Cf. [3, 5.6]) Let p : E - ! X by an n-fold ramified covering map with multiplicity function ~. Then the composite p p* [E, F] -t! [X, F] -! [E, F] is given as follows. Take ['] 2 [E, F], then p*tp['] is represented by the map '0: E -! F given by X '0(e) = ~(e0)e0. p(e0)=p(e) || In the case of an action of a finite group G on E and X = E=G, we have the following consequence. Corollary 2.13. (Cf. [3, 5.7]) If ['] 2 [E, F], then p*tp['] = ['0] 2 [E, F], where X '0(e) = ge . g2G 7 Proof: Just observe that the element ge is repeated in the sum ~(e) = |Ge| times. || Remark 2.14. Considering an action of H on E and a subgroup K H, one has different ramified covering maps as depicted in E FF qK xxxx FqHFF xx FFF --xxx ##F E=K ____________//E=H . qKH One may easily compute several combinations of the functions induced by these covering maps in homotopy sets 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 2.15. 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. 8 The cohomology transfer behaves well with respect to composite ramified covering maps. Proposition 2.16. The following holds: q k fiq k øpOq = øp O øq : Hk(Z; G) -fi!H (Y ; G) -! H (X; G) . Proof: We prove that øpOq = øp O øq : [Z, F] - ! [X, F] for any abelian topological group F. Take w = [h]P2 [Z, F], v = [g] 2 [Y, F], u = [f] 2 [Z,PF], then v = øq(w) if g(y) = q(z)=y (z)h(z), and u = øp(v) if f(x) = p(y)=x~(y)g(y). Hence, X X X X f(x) = ~(y) (z)h(z) = ~(q(z)) (z)h(z) = ,(z)h(z) . p(y)=x q(z)=y pq(z)=x pq(z)=x Therefore, øqøp(u) = øpOq(u). || Corollary 2.17. 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 øpl = løp : Hk(E; G) -! Hk(X; G). Proof: Consider the l-fold ramified covering map q : E - ! E such that q = idE and (e) = l for all e 2 E. Then pl= p O q. Then apply Proposition 2.16. || Remark 2.18. The ln-fold covering map pl obtained from l is a sort of spurious ramified covering, since the multiplicity of p is artificially multipl* *ied by l. It is interesting to remark that the previous result shows that the transfer of this new ramified covering pl remains essentially unchanged. 3 General transfers for n-fold ramified covering maps in cohomology In this section we consider representable contravariant functors h and k, that is, h(-) = [-, H] and k(-) = [-, K], where H and K are spaces, in order to study general transfers. 9 Definition 3.1. An (h, k)-transfer for n-fold ramified covering maps asso- ciates to every n-fold ramified covering map p : E - ! X with multiplicity function ~ : E -! N, a function øp : h(E) -! k(X), with the following two properties. 1. Given a pullback diagram eF (3.2) F *E _____//_E p0|| p|| fflffl| fflffl| Y __F__//_X , of n-fold ramified covering maps, the diagram p h(E) __fi_//_k(X) h(Fe)|| |k(F)| fflffl| |fflffl h(F *E) ____0//_k(Y ) fip commutes. 2. Given f : B -! C, then for the diagram (2.11) the following diagram commutes ß h(Cn x n __n)_fi_C//_k(SP nC) (fnx n1_n)*|| |(SPnf)*| fflffl|_ fflffl| h(Bn x n n) _fißB//_k(SP nB) , Remark 3.3. Observe that the transfers just defined need not be homomor- phisms (even when H and K are H-spaces). Note 3.4. Considering the category Ramcov n whose objects are n-fold ram- ified covering maps and whose morphisms are pullback diagrams, one has two functors, namely E, X : Ramcov n - ! Top such that given a covering map p : E - ! X, E(p) = E and X(p) = X. Then a transfer is a natural transformation h O E -!. k O X (between functors Ramcov n - ! Set), that also is a natural transformation h O (-)n x n __n.-!k O SPn (between functors Top - ! Set). If h = k = Hq(-; G), then by 2.9 and 2.10, we have the following. 10 Proposition 3.5. The transfer øp : h(E) - ! h(X) defined in 2.4 is an (h, h)-transfer. || We have the following classification result. Theorem 3.6. (i)Each class w 2 k(SP nH) determines an (h, k)-transfer øw for n-fold ramified covering maps, and conversely (ii)each (h, k)-transfer ø for n-fold ramified covering maps determines a class wfi2 k(SP nH). Moreover, (iii)the class associated to øw is w, and conversely (iv)the transfer associated to wfiis ø. Proof: (i) Take a class w 2 k(SP nH) and let p : E -! X be an n-fold ramified cov- ering map with multiplicity function ~ : E -! Z. We define øw,p : h(E) -! k(X) as follows. Given [f] 2 h(E), let øw,p[f] be the homotopy class of the composite 'p n SPnf n w X ____//_SPE _____//SPH _____//K . In order to show that øw is natural, consider the pullback diagram (3.2). The element k(F ) O øw,p[f] is given by the homotopy class of the composite __F__// _'p_//_n SPnf//_ n __w__// Y X SP E SP H K . 0 On the other hand, the element øw,p Oh(Fe)[f] is given by the homotopy class of the composite 'p0 n * SPn(fOFe) w Y ____//_SP(F E)N_________________//SPnH9_____//K9 NNN ssss NNNN sssns SPneFNN&&N ss SP f SP nE . By the functoriality of the construction SP n, the triangle commutes; there- fore, we only have to show the commutativity of the following diagram _____F_____// Y X 'p0|| 'p| fflffl| fflffl|| SPn (F *E)_____//SPnE . SPnFe 11 To that end, take y 2 Y and consider the fiber p-1(F (y)) = {e1, . .,.er}. Then 'p O F (y) = , where ei is repeated ~(ei) times. Since p0-1(y) = {(y, e1), . .,.(y, er)} and t* *he multiplicity function of p0 is ~ O eF, we have that 'p0(y) = <(y, e1), . .,.(y, e1), . .,.(y, er), . .,.(y, er)> , where (y, ei) appears ~(ei) times. Therefore, SPnFe O 'p0= , where again ei appears ~(ei) times, and so the diagram commutes. (ii) Let ø be an (h, k)-transfer for n-fold ramified covering maps and consider the map ß : Hn x n __n-! SP nH. As remarked in 2.3, ß is an n-fold ramified covering map. Therefore, we have øi : h(Hn x n __n) -! k(SP nH). Let ff : Hn x n __n-! H be given by ff = ai. Then [ff] 2 h(Hn x n __n). We associate to ø the element wfi= øi [ff] 2 k(SP nH). (iii) Let w 2 k(SP nH), and consider the associated transfer øw . The class in k(SP nH) determined by øw is given by øw,i[ff], where ff : Hn x n __n-! H is given by ff = ai. Therefore, øw,i[ff] is the homotopy class * *of the composite 'ß n n __ SPnff n w SP nH _____//SP(H x n n) ____//_SPH ____//_K . Let a = be an element in SP nH, where al appears il times. Then 'i(a) = <, . .,., . .,.. .,., . .,.r>. _______-z______" _______-z______" _______-z______" i1 i2 ir Therefore, SP nff O 'i(a) = = a , i1 i2 ir so that SP nff O 'i = 1. Hence øw,i[ff] = w. 12 (iv) Finally, given an (h, k)-transfer ø, we have wfi= øi [ff]. In order to show that øwø = ø, consider an n-fold ramified covering map p : E - ! X with multiplicity function ~ : E - ! Z and some element [f] 2 h(E) = [E, H]. We shall prove that øwø,p[f] = øp[f]. For that, consider the following two diagrams e'p n __ n __ fnx n1_n n __ E _____//E x n n E x n n __________//_H x n n p|| |iE| iE || |i=iH| fflffl| fflffl| fflffl| fflffl| X __'p__//_SPnE SP nE ____SPnf_____//SPnH . The one on the left-hand side is a pullback diagram while the one on the right-hand side is like (2.11). Hence, by the two properties of the transfer, we have two commutative diagrams p n __ fiß n h(En x n __n)_fi_//k(SP E) h(Hn x n n) ______//k(SP H) (e'p)*|| ('p)*||(fnx n1_n)*|| (SPnf)*|| fflffl| fflffl| fflffl|_ fflffl| h(E) ____fip___//k(X) h(En x n n) _fißE//_k(SP nE) , and putting the one on the right-hand side on top of the one on the left-hand side, we obtain ß n h(Hn x n __n)_fi__//k(SP H) (e'p)*O(fnx n1_n)*|| |('p)*O(SPnf)*| fflffl| fflffl| h(E) ____fip___//k(X) . If we now chase our element [ff] 2 h(Hn x n __n) defined in the proof of (ii) along the top and right-side of the diagram, we obtain [wfiO SP nf O 'p] = øwø,p[f], while if we chase it along the left-hand and bottom side of the diagram we obtain øp[f]. Thus øwø,p[f] = øp[f], as desired. || As a consequence of Theorem 3.6, we obtain the following. Corollary 3.7. There is a one-to-one correspondence between (h, k)-transfers ø and elements w in k(SP nH). || 13 In the following result we compute w for the cohomology transfer øp defined in 2.4. Proposition 3.8. Let H = F (Sq, G). Then the element wfi2 [SP nH, H] that corresponds to the transfer øp is given by wfi = a1 + . .+.an . Proof: Let ß : Hnx n__n-! SP nH be the n-fold ramified covering map given above.PThe transfer øi : h(Hn x n __n) -! h(SP nH) is such that øi [ff](x) = -1 i(e)=x~(e)ff(e). Thus, if x = , then ß (x) = { | i = 1, . .,.n}. Hence Xn Xn wfi(x) = øi [ff](x) = ff = ai. i=1 i=1 || Definition 3.9. By Theorem 3.6, given representable functors h and k, we can define the set of transfers from h(E) to k(X) for each n-fold ramified covering map p : E - ! X. We denote this set by TnR(h, k). If we assume that the functor k takes values in the category Ab of abelian groups, then we can give TnR(h, k) a group structure as follows. Given oe, ø 2 TnR(h, k) and an n-fold ramified covering map p : E - ! X, we define the transfer oe + ø by (oe + ø)p(a) = oep(a) + øp(a), for every a 2 h(E). Corollary 3.10. Assume that k takes values in Ab. Then the bijection of Corollary 3.7 gives an isomorphism of abelian groups TnR(h, k) ~=k(SP n(H)) Proof: By 3.7, there is a bijection _ : TnR(h, k) - ! k(SP n(H)) given by _(ø) = wfi= øi [ff], as in the proof of 3.6(ii). Then _(oe + ø) = (oe + ø)i[ff] = oei[ff] + øi [ff] = _(oe) + _(ø) . Therefore, _ is an isomorphism. || The following is a nice consequence of this corollary. 14 Proposition 3.11. Let ø be an (h, k)-transfer and assume that there is a commutative diagram Z>>``@ ~0"""" @@~@ """ @@@ " q @ E0 AA__________//_E~ AAA ~~~~ p0AA__A~~p~~~ X of n-fold ramified covering maps, such that q : E0 -! E is surjective. Then the following triangle commutes: q* 0 (3.12) h(E) _______________//h(E ) GG ww GGG wwww fipG##GG --wfip0ww k(X) Proof: By the classification result 3.7, there is an element w = wfi2 k(SP nH) such that for any p : E - ! X and any element [f] 2 h(X) = [X, H], its transfer is given by the composite 'p n SPnf n w øp : X -! SP E -! SP H -! K . Consider the following diagram: SP;nE0K; 'p0vvvv || KKSPn(fOq)KK vvv | KKK vv | K%%K w X HH SPnq|| SPnH99____//_K . HHH | rrrrr 'pHHH | rrr n H##fflffl|SPfrr SP nE The triangle on the right-hand side commutes clearly. The one on the left- hand side commutes too, since X X 'p(x) = ~(e)e , 'p0(x) = ~0(e0)e0, p(e)=x p0(e)=x and thus X X X SP nq'p0(x) = ~0(e0)q(e0) = ~(q(e0))q(e0) = ~(e)e , p0(e0)=x p(q(e0))=x p(e)=x 15 where the last equality follows since q is surjective. Hence, 0 p0 * øp[f] = øw,p[f] = øw,p [f O q] = ø q [f] . || The following theorem tells in some cases about the existence of transfers. Theorem 3.13. Let H* denote singular cohomology with coefficients in Z. Then 8 ><0 if n s > r (s > 0) TnR(Hr, Hs) ~= Z if n s = r >: 0 if n s = r + 1. Proof: By 3.6 and 3.7, we have an isomorphism TnR(Hr, Hs) ~=Hs(SP n(K(Z, r)) . By [2, 6.3.24], for any (r - 1)-connected CW-complex X, the inclusion X ,! SP1 X is an (r + 1)-equivalence. Therefore, SP 1K(Z, r) is (r - 1)-connected, and so ßr(SP 1 K(Z, r)) ~= Z and ßr+1(SP 1 K(Z, r)) = 0. By the Hurewicz theorem, eHi(SP 1 K(Z, r)) = 0 for i < r , Hr(SP 1 K(Z, r)) ~=Z and Hr+1(SP 1 K(Z, r)) = 0 . By the universal coefficients theorem, Hs(SP 1 K(Z, r)) = 0 for s < r , Hr(SP 1 K(Z, r)) ~=Hom (Hr(SP 1 K(Z, r)); Z) ~=Hom (Z, Z) ~=Z . Since Hr(SP 1 K(Z, r)) ~=Z, Ext(Hr(SP 1 K(Z, r)); Z) = 0, and we have that Hr+1(SP 1 K(Z, r)) ~= Hom (Hr(SP 1 K(Z, r)); Z) = 0. By [11], for any CW- complex X, Hs(SP 1 X) ~=Hs(SP nX) for n s, so the result follows. || 16 4 Comparison between transfers for ordinary covering maps and for ramified covering maps In this section we shall compare our classification of transfers for ramified covering maps given in the previous section with the classification of transfers for n-fold ordinary covering maps obtained by Roush [8]. For a description of his result we follow [1]. Definition 4.1. Take again h(-) = [-, H] and k(-) = [-, K] as above. an (h, k)-transfer for n-fold covering maps associates to every n-fold covering map p : E -! X over a paracompact space X a function tp : h(E) -! k(X), which is natural with respect to pullbacks in the same sense of property 1. in the definition of the (h, k)-transfers for n-fold ramified covering maps (3.1). Denote by Tn(h, k) the set of transfers for n-fold ordinary covering maps, and let E n -! B n be the universal principal n-bundle. Then we have the following. Theorem 4.2. (Roush) There is a bijection [E n x n Hn, K] -! Tn(h, k) . || Since the transfers for n-fold ramified covering maps are also natural with respect to pullbacks, as just mentioned above, we have a restriction function r : TnR(h, k) -! Tn(h, k). The following theorem relates both classifications, namely Theorems 4.2 and 3.7. Theorem 4.3. Let æ : E n x n Hn -! SP nH be given by æ = . Then the following diagram commutes. ~= [SP nH, K] oo_____//_TnR(h, k) j*|| r|| fflffl| fflffl| [E n x n Hn, K] _____//Tn(h, k) Proof: Let w : SP nH -! K be a map and øw the transfer for n-fold ramified covering maps associated to it according to Corollary 3.7. Consider æ*[w] = [w O æ] and let p : E - ! X be an n-fold covering map and g : E - ! H. 17 The value of the transfer_tp associated to the class æ*[w], on [g] is defined as_follows. Let q : E - ! X be the principal n-bundle associated to p, i.e., E = {(e1, . .,.en) 2 En | ei 6= ej if i 6= j; and p(e1)_= . . .= p(en)}, and q(e1, . .,.en) = p(e1). There is a free_ n-action on E defined by permuting coordinates, and a homeomorphism fl : E = n -! X given by fl = p(e1). Therefore there is a pullback square ______fi___// E E n | | | | __ fflffl| fflffl| X oo___E = n _____//E n= n ______B n . Then tp[g] 2 k(X) = [X, K] s the class of the composite __ _ n idx ngn n wOj X E = n ____//_E n xffnE __________//_E n x n H ____//_K , where _ = . Now we consider tha foll* *ow- ing diagram: __ _ n idx ngn n wOj E = n _____//E n xffnE __________//_E n x n H _____//K8,8q qqq fl|| j0|| j|| qqwqq fflffl| fflffl| fflffl|qqq X ____'p____//SPnE_______SPng______//SPnH where æ0 = . Since p : E -! X is an n-fold covering map, ~(e) = 1 for all e 2 E, and {e1, . .,.en} is the fiber over p(e1). Therefo* *re, 'p(fl) = . Since æ0_ = , th* *e left- hand side square of the diagram commutes, the middle square as well as the triangle are clearly also commutative. But the class of the composite w OSP ng O'p is fl(øw )p[g] = øw,p[g]. Hence fl(øw )p[g] = tp[g] and thus fl(øw* * ) = t. || 5 Transfers for ramified covering maps in coho- mology In this section we shall consider families of (h, k)-transfers for n-fold ramif* *ied covering maps for all n. Assume that H and K are topological abelian groups, 18 in order to have group structures in the homotopy sets [X, H] and [X, K] for all X. Before stating the relevant definition, consider an n-fold ramified covering map p : E -! X with multiplicity_function_~ : E -! Z. Recall the (n + 1)- fold ramified covering map __p: E = E t X -! X given in Remark 2.2. Definition 5.1. An (h, k)-transfer ø for ramified covering maps consists of an (h, k)-transfer øn for n-fold ramified covering maps, for each n = 1, 2, 3, . .,.such that for each n-ramified covering map p : E -! X the triangles in the following diagram commute: i*2 i*1 (5.2) h(X) oo___h(E t X) _____//h(E) KK ss KKK | _psss fiXKK%%KKKfin+1fflffl||fipnyysssss k(X) , where i1 and i2 are the canonical inclusions, and øX = øidX1, which is just given by a natural transformation h -.! k. _p Observe that since i*1is an epimorphism, øpnis determined by øn+1 for any n-fold ramified covering map p : E - ! X. Therefore, we have an inverse system . . .-! TnR+1(h, k) -! TnR(h, k) -! . .-.! T1R(h, k) = Nat(h, k) , where Nat (h, k) denotes the natural transformations from h to k. Thus, a transfer for ramified covering maps is an element in limnTnR(h, k) = T1R(h, k). On the other hand, we have another inverse system * n . .-.! k(SP n+1H) -i! k(SP H) -! . .-.! k(H) , where i : SP nH ,! SP n+1H is the canonical inclusion given by 7! . By Corollary 3.10, we have the following. Theorem 5.3. There is an isomorphism T1R(h, k) -! limnk(SP nH). More precisely, the diagram ~= n+1 TnR+1(h, k)____//k(SP H) | | * | |i fflffl| fflffl| TnR(h, k)__~=__//k(SP nH) commutes for all n. 19 Proof: Let ø be an (h, k)-transfer for ramified covering maps. We have to show that for each n, i*(wfin+1) = wfin. Recall that wfin= øinn(ffn), where ßn : Hn x n __n-! SP nH is the canonical ramified covering map, and ffn : Hn x n __n-! H is given by 7! ai. Let p : E -! SP nH be the (n + 1)-fold ramified covering map obtained by taking the pullback of ßn+1 over i : SP nH ,! SP n+1H. Thus we have a commutative square ______ j* h(Hn+1 x n+1 n + 1) _______//_h(E) fißn+1n+1|| fipn+1|| fflffl| fflffl| k(SP n+1H) ____i*___//_k(SP nH) , where j is the induced inclusion. By the universal property of the pullback, there is a (unique) surjective map q : (Hn x n __n) t SPn H -! E such that p O q|Hnx n_n = ßn (and p O q|SPnH = idSPnH). Thus, combining (3.12) and the right-hand side of (5.2) for this case, we have a commutative diagram q* n __ n i*1 n __ h(E) ____//_RRh((H x n n) t SP H)__//_h(H x n n) RRR 0 jjjjj RRRR |fißn jjjjj fipn+1RRR((RRffln+1ffl||fißnnttjjjjj k(SP nH) . Therefore, i*(wfin+1) = øpn+1j*(ffn+1) = øinni*1q*j*(ffn+1) = øinn(ffn) = wfin, since i*1q*j*(ffn+1) = (j O q O i1)*(ffn+1) = ffn, as_one_easily verifies after observing that j O q O i1 : Hn x n __n-! Hn+1 x n+1 n + 1 is the canonical inclusion. || Example 5.4. The (h, h)-transfers øpn= tp given in Definition 2.4 for each n determine_an_(h, h)-transfer for ramified covering maps, since for any [eff] 2 h(E ) = [E , H], its images on both sides of Diagram (5.2) are given by_[eff1] = [eff|E ] 2 h(E) = [E, H] and [eff2] = [eff|X ] 2 h(X) = [X, H].PThen øpn+1[eff]* * = [ff], øpn[eff1] = [ff1], and øX [eff2] = [ff2], where ff(x) = p(e)=x~(e)eff(e* *)+fef(x) = P _p p p(e)=x~(e)eff1(e)+eff2(x). Thus øn+1[eff] = øn[eff1]+øX [eff2], and hence Dia* *gram (5.2) commutes in this case. Assume in what follows that ø is an (h, h)-transfer for ramified covering maps given by an element [w] 2 limn[SP nH, H], h = [-, H]. Supposing that 20 (H, 0) is a well-pointed space, then the inclusion in : SP n-1H ,! SP nH is a cofibration and we may thus assume that [w] is given by a family of maps wn : SP nH - ! H such that wn-1 = wn O in. If we further assume that øX = 1h(X). Then we have that w1 ' idH , and we may suppose from the start that w1 = idH. We thus have that the maps wn determine a map w : SP 1H -! H that has the property that w|SPnH = wn. In particular, it has the property that w|H = idH. We have the following. Lemma 5.5. Let H have the homotopy type of a connected CW-complex. If there is a map w : SP 1H - ! H such that w|H = idH, then H has the Q homotopy type of a weak product en 0K(ßn(H), n), of Eilenberg-Mac Lane spaces. Proof: The homomorphism w* : ßn(SP 1 H) -! ßn(H) splits i* : ßn(H) -! ßn(SP 1 H) for all n. By the Dold-Thom theorem (see [2]), ßn(SP 1 H) ~= eHn(H; Z) and under this isomorphism, i* corresponds to the Hurewicz ho- momorphism. Thus, the Hurewicz homomorphism is a split mono and hence by a theorem of Moore (see Theorem 5.1 in [3]) we have the result. || Hence, by the previous lemma and Theorem 5.3, we have the following consequence. Theorem 5.6. Let H have the homotopy type of a connected CW-complex. There is an (h, h)-transfer ø for ramified covering maps such that øidX1= 1h(X) if and only if H has the homotopy type of a weak product Yf K(ßn(H), n) , n 0 of Eilenberg-Mac Lane spaces. || Let now H be an abelian H-group with (strict) neutral element 0 2 H, and let its multiplication map be : H x H -! H. We can define a map n : Hn -! H by 0 = 0, the constant map with value 0, 1 = idH , and inductively n(a1, . .,.an) = 2( n-1(a1, . .,.an-1), an). Then we define multiplication by n in H as the map nO n : H -! H, where n : H -! Hn is the diagonal map. Proposition 5.7. Let ø be a transfer for ramified covering maps classified by a family wn : SP nH - ! H and let H be an abelian H-group with mul- tiplication given by , n 2 N. Assume, moreover, that øidX1= 1h(X). Then 21 øpn: h(E) -! h(X) is a homomorphism for every n-fold ramified covering map p : E - ! X if and only if wn O ßn ' n, where ßn : Hn -! SP nH is the identification. Proof: Take [g], [g0] 2 h(E) = [E, H], then [g] + [g0] = [ O (g x g0) O E ], where E : E -! E x E is the diagonal map. Consider the diagram (5.8) SPn E n SPn(gxg0) n SPn n SP:nE:_____//SP(EOxOE) _________//SP(H xOH)O _____//SPHK 'puuuu | | KKwnKK uuu | | KK%% X J || || 9H9. JJJJ | | rrrr X J$$ fflffl| fflffl| rr X x X 'px'p//_SPnE x SPnSEPngxSPng0//_SPnH x SPnwHnxwn//_H x H Since the two left subdiagrams are always strict commutative, the full di- agram is (homotopy) commutative if and only if the right subdiagram is (homotopy) commutative. To see the necessity of this commutativity, just consider the trivial n-fold covering map p = idH : H -! H and g = g0= id. Hence, øn is a homomorphism if and only if the diagram n (5.9) SP nH x SPn H ______SPn(H x H) SP__//_SPnH wnxwn || wn|| fflffl| fflffl| H x H ____________________________//H is homotopy commutative. Since w1 ' idH, because øidX1= 1h(X), in the case n = 2, the diagram means that we have two operations on H with a common zero, namely and w2 O ß2, that are mutually distributive up to homotopy. By following the proof of Lemma [2, 2.10.10] up to homotopy, one can show that both operations are homotopic, that is, w2 O ß2 ' 2 (Observe that this means that factors through SP 2H up to homotopy). In the case n = 1, take the trivial 1-fold covering map p = idH : H -! H again and take g = id, g0 = 0. Then the commutativity of Diagram (5.8) shows that w1 ' id, since O (id, 0) = id. Further, using the definition of n and the commutativity up to homotopy of (5.9) one may prove inductively that wn O ßn ' n. || According to Proposition 3.8, we obtain the following. 22 Corollary 5.10. If an (h, h)-transfer for ramified covering maps ø such that øidX1= 1h(X), yields a homomorphism for every n-fold ramified covering and every n, then it is unique (it is namely, the one defined in 2.4), and it has thus the property that for every n-fold ramified covering p : E - ! X, the composite øpnO p* is multiplication by n. Proof: Since wn O ßn ' n, it follows that wn is homotopic to the map 7- ! a1 + . .+.an . Hence by 3.8, ø is the transfer defined in 2.4 for H. Moreover øpnp*[g] = n[g] as follows from the commutativity of the following diagram 'p n SPn(gOp) n wn X|_____//FFSPE__________//SPH6_____//H6,mGG___________ | FFFF | n mmmmmm __________________________________ | n FF##FSPfpflffl||SPngmmmmmm_________________________ ______________ g || SP nX ________________________________________* *___ | ___________________________________ | ______________________________________ | _______________________________________________ fflffl| ______n____________________________________________* *______________________________________ H ____________________________________________________________* *_________________________________________________________________@ where the top row provides øpnp*[g], since wn ' n. Note that this follows also explicitly from the very Definition 2.4, as shown in Proposition 2.7. || From 5.7, we have the following. Lemma 5.11. Let H be a topological abelian group with multiplication given by : H x H - ! H. Let moreover ø be an (h, h)-transfer for ramified covering maps such that øidX1= 1h(X) and that yields a homomorphism for every n-fold ramified covering and every n. If wn : SP nH - ! H are the classifying maps for ø, then for all n, n0, the following diagram commutes up to homotopy: 0 wnxwn0//_ SP nH x SPn H H x H | | | | fflffl|0 fflffl| SP n+n H __wn+n0__//_H . Proof: Since wn O ßn ' n, and O ( n x n0) = n+n0, as follows from the associativity of , the desired commutativity follows. || 23 We obtain immediately the following. Proposition 5.12. Let p : E -! X, resp. p0: E0 -! X, be an n-fold, resp. n0-fold, ramified covering map and take the (n+n0)-fold ramified covering map (p, p0) : EtE0 -! X defined py p and p0. If ø is an (h, h)-transfer for ramified covering maps such that øidX1= 1h(X) and that yields a homomorphism for every n-fold ramified covering and every n, then 0) p p0 0 ø(p,pn+n0[g] = øn[f] + øn0[f ] , where [g] corresponds to ([f], [f0]) under the obvious isomorphism [E t E0, H] ~=[E, H] [E0, H] . Proof: The following diagram commutes: 0 0 0 0 __SPnfxSPn_f_// n+n0 SP nE7x7SPn E SP HJ ('p,'0p)ooo|oo | Jwn+wn0JJJ ooo | | JJJ oo | | %% X NN | | H:,: NNN | | tt N0NNN | | ttt '(p,p )N''0fflffl| fflffl|wn+n0ttt SP n+n (E t E0)________0____//SPn+n0H SPn+n g where wn + wn0 = O (wn x wn0). Hence, the right-hand-side subdiagram commutes by Lemma 5.11. The other two subdiagrams commute obviously, where the vertical arrows are clear. 0 Following the diagram along the bottom yields ø(p,pn)+n0[g], while doing it 0 0 along the top yields øpn[f] + øpn0[f ]. || In the same situation as above and identifying g with (f, f0), we obtain the following. Corollary 5.13. A family {øn}n=1,2,...such that each øn is an (h, h)-transfer for n-fold ramified covering maps and such that øidX1= 1h(X), determines an (h, h)-transfer for ramified covering maps if and only if 0) 0 p p0 0 ø(p,pn+n0[(f, f )] = øn[f] + øn0[f ] . || 24 In what follows, we shall show that the transfers for ramified cover- ing maps in singular cohomology, i.e., the elements of limn TnR(h, k), where h(X) = Hq(X; G) and k(X) = Hq(X; G0), are determined by the transfers for 2-fold ramified covering maps. Theorem 5.14. The restriction function r : TnR+1(Hq(-; G), Hq(-; G0)) -! TnR(Hq(-; G), Hq(-; G0)) is an isomorphism for n 2 (q > 0). Proof: Since h = Hq(-; G), H = K(G; q), which is a CW-complex. By [12], SPnH is also a CW-complex and a subcomplex of SP n+1H. Therefore, we have a (H-coexact) cofibration sequence i n+1 j n+1 n SPnH ,! SP H i SP H=SP H . n+1 Let us denote by fSP H the reduced symmetric product of H, i.e., the quo- tient of the action of n+1 on the smash product H ^ . .^.H (n + 1 factors). n+1 Clearly, SP n+1H=SP nH fSP H. Using the exact cohomology sequence of the cofibration sequence above for the theory k(X) = Hq(X; G0) gives us an exact sequence j* n+1 i* q n 0 ffi n+1 Heq(fSPn+1H; G0)__//Hq(SP H; G0)_//_H (SP H; G )_//eHq+1(fSP H; G0) , and Theorem 5.3, gives us the commutative diagram * q n 0 Hq(SP n+1H;OG0)O _______i_______//_H (SPOOH; G ) ~=|| |~=| fflffl| fflffl| TnR+1(Hq(-; G), Hq(-; G0)) _r__//_TnR(Hq(-; G), Hq(-; G0)) . n+1 Since H = K(G; q) is (q - 1)-connected, by [12], fSP H is (2n - 2 + q)- connected. Therefore, by the Hurewicz isomorphism theorem and the univer- n+1 n+1 sal coefficient theorem, we have that eHq(fSP H; G0) and eHq+1(fSP H; G0) are both zero, provided that n 2. Hence, r is an isomorphism for n 2. || 25 Remark 5.15. From Proposition 2.8 it follows that in the case that H = K(Zp, q) = K, if p is an odd prime, then w2 is characterized uniquely up to homotopy by the equation w2 O ß2 ' 2 proved in Proposition 5.7. In the case of theories other than the ordinary ones (given by Eilenberg- Mac Lane spaces), there are nontrivial transfers. The following is an interest- ing case. Example 5.16. We analyze transfers for vector bundles. Let H be BU (1) and K be BU (n). We denote by VectkC(X) the set of isomorphism classes of numerable complex k-dimensional vector bundles over X. By 3.7, given an n-fold ramified covering map p : E - ! X, there is a bijection between transfers tp : Vect1C(E) -! VectnC(X) and elements in [SP nBU (1), BU (n)] ~= VectnC(SP nBU (1)). Notice that BU (1) is a topological abelian group of which one can give a model by defining BU (1) = F (S1, U(1)) (see [7]). This group structure is given in terms of line bundles by the tensor product. Since BU (1) is an Eilenberg-Mac Lane space of type K(Z, 2), this group structure also corre- sponds to the group structure in H2(X; Z), where one maps each line bundle to its first Chern class. Consider the map : BU (1)n -! BU (1) defined by the product above, and which corresponds to the bundle fl1 . . .fl1, where fl1 is the universal line bundle. Since BU (1) is abelian, this map defines a map __: SP nBU (1) -! BU (1). Now let æ : BU (1) -! BU (n) be the classifying map of the Whitney sum fl1 . . .fl1 of n copies of fl1. Then the homotopy class of æ O __defines a transfer as we mentioned above. To see that this transfer is not trivial, consider the diagonal map d : BU (1) -! BU (1)n. Denote by ~ the tensor product fl1 . . .fl1 of n copies of fl1. Then the map æ O __O p O d = æ O O* * d classifies the bundle ~ . . .~ (n copies). By the comments above, fl1 7! c1(fl1) yields an isomorphism, and H2(BU (1); Z) ~= Z, therefore c1(~) = nc1(fl1) 6= 0. Hence c1(~ . . .~) = c1(~) + . .+.c1(~) = n2c1(fl1) 6= 0, and so the transfer defined is not trivial. Example 5.17. Now, we analyze transfers for principal bundles. Let ß : P - ! E be a principal -bundle, where is a topological abelian group, and let {Uff| ff 2 I} be a trivializing open cover of E for ß. Let {gff fi: Uff\ Ufi-! | ff, fi 2 I} 26 be a set of transition functions for the bundle, so that for every x 2 Uff\ Ufi\ Uflthey satisfy gff fi(x)gfi(flx) = gff fl(x) . The nth product bundle ßn : P n- ! En with fiber n is a principal bun- dle with trivializing open cover {Uff1x . .x.Uffn| (ff1, . .,.ffn) 2 In} and transition functions g(ff1,...,ffn) (fi1,...,fin):(Uff1x . .x.Uffn) \ (Ufi1x . .x.Ufin) = gff1fi1x...xgffnfin (Uff1\ Ufi1) x . .x.(Uffn\ Ufin)___________//_ n . Let q : En -! SP nE be the quotient map and define U(ff1,...,ffn)= q(Uff1x . .x.Uffn), which is an open set since q is an open map. Given two multi- indexes (ff1, . .,.ffn), (fi1, . .,.fin) 2 In, define g0(ff1,...,ffn) (fi1,...,fin): U(ff1,...,ffn)\ U(fi1,...,fin)-! by g(ff1,...,ffn)(fi1,...,fin) (Uff1x . .x.Uffn) \ (Ufi1x . .x.Ufin)________________//_ n | q||| || fflffl| eg fflffl| U(ff1,...,ffn)\ U(fi1,...,fin)______________//SPn X X X XX | X X X X | g0(ff1,...,ffn)(fi1,...,fin),,XXXXXXfflff* *l| , where the restriction of q on the left-hand side is a surjective open map, thus an identification. Hence egis well defined. The map SP n -! given by multiplying the n elements of is well defined too, since is abelian, and its composite with egyields the desired map. In order to see that these maps g0(ff1,...,ffn) (fi1,...,fin)are transition functions, we check the releva* *nt relation. Take x 2 U(ff1,...,ffn)\ U(fi1,...,fin)\ U(fl1,...,fln). Thus x = q(x1, . .,.xn* *), where xi 2 Uffi\ Ufii\ Ufli, i = 1, . .,.n. These elements are well defined, since we are taking ordered multi-indexes (permutations in the multi-indexes yield different transition functions, even with the same open sets in SP nE). We 27 have g0(ff1,...,ffn) (fi1,...,fin)(x)g0(fi1,...,fin)((fl1,...,fln)x)= = gff1fi1(x1)gfi1fl1(x1) . .g.ffnfin(xn)gfinfln(x* *n) = gff1fl1(x1) . .g.ffnfln(xn) = g0(ff1,...,ffn) (fl1,...,fln)(x) . Thus these maps are, indeed, transition functions for a new principal - bundle ß0: E0 -! SP nE. Assume now that p : E - ! X is an n-fold ramified covering map and take 'p : X - ! SP nE. If ß : P - ! E is principal -bundle as above, then define the bundle øp(ß) : øp(P ) -! X to be the pullback of ß0: E0 -! SP nE over 'p. If the given principal -bundle ß : P - ! E is numerable and is classified by a map fi : E - ! B , then øp(ß) : øp(P ) - ! X is classified by the composite O SP nfi O 'p : X - ! B , where : SP nB - ! B is given by the multiplication in B . Indeed, if is an abelian group, one can give a model of B that is also an abelian (topological) group. For instance, we may take B = F (S1, ) as the McCord topological group (see [7, 9.17]). If we denote by Prin (Y ) the set of (equivalence classes of) numerable principal -bundles over Y , then the transfer of the n-fold ramified covering map p : E -! X is a function øp : Prin (E) -! Prin (X) or, equivalently, øp : [E, B ] -! [X, B ] . According to the classification result 3.7, this transfer corresponds precisely to the element [ ] 2 [SP nB , B ]. A modified version of the previous example is the following. Example 5.18. Let ß : , - ! E be a (numerable) k-dimensional F-vector bundle (F = R or C), and let {gff fi: Uff\ Ufi-! GL (k, F) | ff, fi 2 I} be a set of transition functions for this bundle. Consider the determinant bundle det ß : det , - ! E, whose transition functions are given by {det Ogff fi: Uff\ Ufi-! F* = GL (1, F) | ff, fi 2 I} , 28 where det : GL (k, F) -! F* is the determinant function. Applying the construction given in Example 5.17 for = F*, we obtain a line bundle ß0: ,0 -! SP nE for any n. As in 5.17, if p : E -! X is an n-fold ramified covering map with 'p : X - ! SP nE, then the pullback of ß0 over 'p defines a bundle øpn(ß) : øpn(,) -! X. Thus we have a transfer øpn: VectFk(E) = [E, BGL (k, F)] -! [X, BF*] = VectF1(X) . To compute the group of transfers of the previous example in the complex case, we need the following. Lemma 5.19. Let detk : U(k) -! U(1) be the determinant function. Then Bdetk : BU (k) -! BU (1) is such that [Bdet ] = c1(flk), the first Chern class of the universal bundle, thus it is a generator of H2(BU (k)) ~=Z. Proof: Take BU (k) = F (S1, U(k)). Then Bdetk : F (S1, U(k)) -! F (S1, U(1)) is given by Bdet k(u) = detkOu. Since BU (1) = F (S1, U(1)) is a K(Z, 2), we take it to represent the second cohomology groups. If k = 1, then det1 : U(1) -! U(1) is the identity, hence [Bdet 1] = [id] 2 H2(BU (1)) = [BU (1), BU (1)] , which is the generator, since we are dealing with Eilenberg-Mac Lane spaces, and we have that ~= [BU (1), BU (1)] -! Hom (ß2(BU (1)), ß2(BU (1))) ~=Hom (Z, Z) ~=Z . By induction, we assume that the homotopy class of Bdet k: BU (k) -! * BU (1) is c1(flk), and consider H2(BU (k + 1)) Bi-!H2(BU (k)), where i is the inclusion map. By naturality and stability of the Chern classes, Bi*c1(flk+1) = c1(flk). Since these classes are the generators, Bi*`is'an isomorphism. Recall- A 0 ing that i : U(k) -! U(k + 1) maps a matrix A to , one has that 0 1 Bi*[Bdet k+1] = [Bdet k], since ` ' u(x) 0 Bdet k+1O Bi : x 7- ! detk+1iu(x) = detk+1 = detku(x) . 0 1 Therefore, [Bdet k+1] = c1(flk+1) and thus it is a generator. || 29 We have the following result. Theorem 5.20. T R(Vect Ck(-), VectC1(-)) ~=Z and the family of transfers øpn constructed in Example 5.19 constitute a generator. Proof: First we analyze the situation for the transfers for n-fold ramified covering maps. By Corollary 3.10, TnR(Vect Ck(-), VectC1(-)) ~=H2(SP nBGL (k, C); Z) . Without losing generality, we may write U(k) instead of GL (k, C). Making use of the fibration U(k - 1) ,! U(k) - ! S2k-1, one easily shows that ß1(U(k)) ~=Z for k 1. With this and the same arguments used in the proof of 3.13, one has H2(SP nBGL (k, C); Z) ~=Z . Now, consider the inverse system (5.21) . .-.! H2(SP 3BU (k)) -! H2(SP 2BU (k)) -! H2(BU (k)) . To show that_all arrows are isomorphisms, take the cofiber sequence SPn X ,! n+1 n+1 n SPn+1X i SP X = SP X=SP X. By [12] we have that if X is (l - 1)- ___n+1 connected, then SP X is (2n + l - 2)-connected. Therefore, since BU (k) is 1-connected, in the exact sequence ~= ___n+1 He2(___SPn+1BU(k))_//eH2(SP n+1BU (k))//_eH2(SP nBU (k))//_eH3(SP BU (k)) || || 0 0 the middle arrow is an isomorphism if n 2. In order to see that the last arrow on the right in the inverse system (5.21) is also an isomorphism, we do the following. Take BU (k) = F (S1, U(k)), and take as base point * the function given by *(s) = 1, the identity matrix, for all s 2 S1. In particu- lar, BU (1) is a topological abelian group. On the other hand the inclusions SPnBU (k) ,! SP n+1BU (k) are given by 7! . By Lemma 5.19, the generator of H2(BU (k)) = [BU (k), BU (1)] as an infinite cyclic group is given by [Bdet ]; on the other hand, the homotopy classes of the maps fin given by the diagrams (Bdet)n n BU (k)n ______//BU(1) ____//_BU(1)44ii iii | iiiiii | iiiiifini fflffl|iii SP nBU (k) 30 seen as elements in He2(SP nBU (k)), where the top arrow on the right-hand side is given by the abelian multiplication in BU (1), obviously map to each other in the inverse system (5.21). In particular, [fi2] 7! [fi1] = [Bdet ]. So* *, the last arrow on the right of the inverse system is surjective, and thus it is an isomorphism. Hence, all arrows are isomorphisms and the elements [fin] are generators of the infinite cyclic groups, and since each øpncorresponds to [fin* *], the family øp = {øpn} is a transfer for ramified covering maps. Consequently, T R(Vect Ck(-), VectC1(-)) ~=limn H2(SP nBU (k)) ~=Z, and øp is the generator. || 6 Transfers in 1-dimensional integral cohomology We consider (H1(-; Z), H1(-; Z))-transfers for n-fold ramified covering maps as well as for n-fold covering maps. We denote by n sZ the wreath product of n and Z, i.e., the semidirect product of n with Zn, where n acts on Zn by permuting the summands. Therefore, the product in n sZ is given by (oe, a1, . .,.an) . (ø, b1, . .,.bn) = (oeø, afi(1)+ b1, . .,.afi(n)+ bn). Lemma 6.1. The following hold. (a) B( n sZ) = E n x n (R=Z)n. (b) Let f : n sZ -! Z be the homomorphism defined by f(oe, a1, . .,.an) = a1 + . .+.an. Then Bf : B( n sZ) -! BZ is given by _ _ _ _ ' : E n x n (R=Z)n -! R=Z , where ' = t1+ . .+.tn. Proof: (a) Consider the space E n x Rn. The space is contractible and has a free action of n sZ given by (y, t1, . .,.tn) . (oe, a1, . .,.an) = (y . oe, tff(1)+ a1, . .,.tff(n)+ * *an) . Therefore, B( n sZ) = (E n x Rn)=( n sZ). Now consider the following diagram idxqn n E n x Rn _________________//_E n x (R=Z) | | | | fflffl| fflffl| (E n x Rn)=( n sZ) ` ` ` `` `//E n x n (R=Z)n . 31 Since the quotient map q : R -! R=Z is a covering map, so is also idxqn; in particular, it is a quotient map as are also the two vertical maps. Thus they clearly define a homeomorphism (E_n x Rn)=(_n sZ) -! E n x n (R=Z)n given by 7! . (b) Consider the action n sZ x Z -! Z given by (oe, a1, . .,.an) . k = f(oe, a1, . .,.an) + k = a1 + . .a.n+ k . By part (a), we have a principal n sZ-bundle_p_: E n x Rn -! E n x n (R=Z)n, where p(y, t1, . .,.tn) = . Then Bf classifies the asso* *ci- ated principal Z-bundle __p: (E n x Rn) x ns ZZ -! E n x n (R=Z)n. Now consider the following diagram _ (E n x Rn) x ns ZZ ______//_R _p| q| | | fflffl| fflffl| E n x n (R=Z)n ___'__//_R=Z , where _<(y, t1, . .,.tn), k> = t1 + . .+.tn + k. Clearly this is a morphism of principal Z-bundles; therefore, Bf ' '. || Lemma 6.2. H1(E n x n K(Z, 1)n; Z) ~=Z. Proof: Since K(Z, 1) = BZ and by 6.1, E n x n BZn = B( n sZ), we have that H1(E n x n K(Z, 1)n; Z) ~=H1(B( n sZ); Z) ~=Hom ( n sZ, Z). Let ' : Zn ,! n sZ be the inclusion given by '(a1, . .,.an) = (1, a1, . .,.an), and consider '* : Hom ( n sZ, Z) -! Hom (Zn, Z). Let F : n sZ -! Z be a homomorphism and assume that '*(F ) = 0. Then F (1, a1, . .,.an) = 0 for all (a1, . .,.an) 2 Zn. Since any element (oe, a1, . .,.an) 2 n sZ can be written as (oe, a1, . .,.an) = (oe, 0, . .,.0) . (1, a1, . .,.an), we have* * that F (oe, a1, . .,.an) = F (oe, 0, . .,.0) + F (1, a1, . .,.an) = F (oe, 0, . .,.0* *). But oe is an element of n, that is a finite subgroup of n sZ and the codomain is free, hence F (oe, 0, . .,.0) = 0. Therefore, F = 0, so that '* is a monomor- phism. Let ei be the element in Zn whose coordinates are all zero, except the ith one that is equal to 1. Let ø 2 n be the permutation given by ø(i) = j and ø(k) = k for k 6= i, j. Then (1, ei) . (ø, 0) = (ø, ej) = (ø, 0) . (1, ej). Hen* *ce F ((1, ei) . (ø, 0)) = F ((ø, 0) . (1, ej)). But F ((1, ei) . (ø, 0)) = F (1, e* *i) and F ((ø, 0) . (1, ej)) = F (1, ej). Therefore, F (1, ei) = F (1, ej). Since we ha* *ve 32 ~= an isomorphism _ : Hom (Zn, Z) -! Zn, given by _(f) = (f(e1), . .,.f(en)), then im(_ O'*) is the diagonal subgroup in Zn, that is isomorphic to Z. More- over, the canonical element in Hom ( n sZ, Z) given by (oe, a1, . .,.an) = a1 + . .+.an is a generator because _'*( ) = ('*( )(e1), . .,.'*( )(en)) = (1, . .,.1). || As a consequence of Lemma 6.2, we have that the (H1(-; Z), H1(-; Z))- transfers for n-fold ramified covering maps are exactly the same as the (H1(-; Z), H1(-; Z))-transfers for n-fold ordinary covering maps. Moreover, there is exactly one transfer for each integer. In other words, we have the following. Theorem 6.3. The restriction r : TnR(H1(-; Z), H1(-; Z)) -! Tn(H1(-; Z), H1(-; Z)) is an isomorphism, and both groups are isomorphic to Z. Proof: By 4.3, we have a commutative diagram ~= 1 n TnR(H1(-; Z), H1(-; Z)) ________//_H (SP (R=Z); Z) r || |j*| fflffl| fflffl| Tn(H1(-; Z), H1(-; Z)) __~=_//H1(E n x n (R=Z)n; Z) . By [2, 5.2.23], the canonical inclusion j : R=Z ,! SP n(R=Z) is_a homotopy_ equivalence._Let_w : SP n(R=Z) -! R=Z be the map defined by w = t1+. .+.tn. Then j*[w] = [wOj] = [id], which is the generator of H1(R=Z); Z). Therefore, H1(SP n(R=Z); Z) ~=Z, with generator given by [w]. By Lemma 6.1(a), H1(E n x n (R=Z)n; Z) ~=H1(B( n sZ); Z) = [B( n sZ), BZ] . By [2, 6.4.6], [B( n sZ), BZ] ~=Hom (ß1(B( n sZ)), ß1(BZ)) = Hom ( n sZ, Z) , and by Lemma 6.2, Hom ( n sZ, Z) ~=Z, with a generator f : n sZ -! Z defined by f(oe, a1, . .,.an) = a1+ . .+.an. Therefore, by the naturality of the homotopy equivalence BG ' G for any discrete group G, the generator of H1(B( n sZ); Z) is given_by Bf._By Lemma_6.1(b),_Bf_' '. Since_æ*[w] = [w O æ], and wæ = w = t1 + . .+.tn, w O æ = '. Therefore, æ* is an isomorphism and therewith r is also an isomorphism. || 33 As an immediate consequence we have the following. Corollary 6.4. There is an isomorphism TnR(H1(-; Z), H1(-; Z)) - ! Z. The canonical transfer ø, as given in 2.4, corresponds to 1 2 Z. For any other integer k the corresponding transfer is kø given by (kø)pn(j) = k(øpn(j)) for a* *ny n-fold ramified covering map p : E - ! X and any element j 2 H1(E; Z). || References [1]J. F. Adams, Infinite Loop Spaces, Annals of Math. Studies No. 90, Princeton Univ. Press, Princeton, NJ 1978 [2]M. Aguilar, S. Gitler, C. Prieto, Algebraic Topology from a Ho- motopical Viewpoint, Universitexts, Springer-Verlag, New York Berlin Heidelberg 2002 [3]M. Aguilar, C. Prieto, Transfers for ramified coverings in homology and cohomology, 2003 (to appear). [4]A. Dold, Ramified coverings, orbit projections and symmetric powers, Math. Proc. Camb. Phil. Soc. 99 (1986), 65-72 [5]P. J. Huber, Homotopical cohomology and ~Cech cohomology, Math. Annalen 144 (1961), 73-76. [6]J. P. May, A Concise Course in Algebraic Topology, Chicago Lecture Notes in Mathematics, The University of Chicago Press, Chicago and London 1999. [7]M.C. McCord, Classifying spaces and infinite symmetric products, Trans. Amer. Math. Soc. 146 (1969), 273-298 [8]F. W. Roush, Transfer in Generalized Cohomology Theories, Pure and Applied Mathematics, Akad'emiai Kiad'o, Budapest 1999 [9]L. Smith, Transfer and ramified coverings, Math. Proc. Camb. Phil. Soc. 93 (1983), 485-493 [10]E. H. Spanier, Function spaces and duality, Ann. of Math. 70 (1959), 338-378 [11]N. Steenrod, Cohomology operations, and obstructions to extending continuous functions, Advances in Math. 8 (1972), 371-416 34 [12]P. J. Welcher, Symmetric products and the stable Hurewicz homo- morphism, Illinois J. Math. 24 (1980), 527-544 January 13, 2004 35