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 nfold 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 nfold 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
Email 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 Hspaces). 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 groupvalued. 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 EilenbergMac 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 EilenbergMac 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 onetoone
correspondence between (h, k)transfers for nfold ramified covering maps
and elements in k(SP nH). We prove further that there are nontrivial trans
fers for nfold 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 nfold ramified covering maps for all n and give their
classification. Namely, we prove that there is a onetoone 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 2fold 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 nfold ramified cover
ing maps in 1cohomology is isomorphic to the group of transfers for ordinary
covering maps, and both are isomorphic to Z.
2 Transfers for nfold 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.Yz____"= 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+...+mk1+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 nfold 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 p1(x) are finite (discrete), x 2 X.
P
(ii)For each x 2 X, e2p1(x)~(e) = n.
P
(iii)The map 'p : X ! SP nE given by 'p(x) = e2p1(x)~(e)e is contin
uous.
Remark 2.2. Given an nfold ramified covering map p : E  ! X with
multiplicity function ~, one can construct an nfold 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 nfold ramified covering map p0: E0 ! X=A, where E0 = E=p1A,
p0 is the map between quotients and the multiplicity function ~0 coincides
with ~ off p1A and is extended by setting ~0(*) = n, if * is the base point
onto which p1A collapses. __
__ Another useful construction is the following. Let E = E t X and __p:
E  ! X be such that __pE = p and __pX = 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 nfold 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 Gfold 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 nfold 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 nfold 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 CWcomplexes, 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 nfold 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 = p1A, 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
fflfflØfflffl" fflffl
A _____//_X___////_X=A .
Then one has cohomology transfers for each covering map. In particular, if A
is a subcomplex of a CWcomplex 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 nfold 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 nfold 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 nfold 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 nfold 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 nfold 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 mnfold ramified covering map, with
multiplicity function , : Z ! N given by ,(z) = (z)~(q(z)). In order
to verify that this composite is indeed an mnfold 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 nfold ramified covering map p : E  ! X with
multiplicity function ~ and an integer l, there is an lnfold 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 lfold 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 lnfold 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 nfold 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 nfold ramified covering maps asso
ciates to every nfold 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 nfold 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 Hspaces).
Note 3.4. Considering the category Ramcov n whose objects are nfold 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 nfold
ramified covering maps, and conversely
(ii)each (h, k)transfer ø for nfold 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 nfold 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 p1(F (y)) = {e1, . .,.er}.
Then
'p O F (y) = ,
where ei is repeated ~(ei) times. Since p01(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 nfold ramified covering maps and consider
the map ß : Hn x n __n! SP nH. As remarked in 2.3, ß is an nfold 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 nfold 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 lefthand side is a pullback diagram while the one on the
righthand 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 righthand side on top of the one on the lefthand
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 rightside of the diagram, we obtain [wfiO SP nf O 'p] =
øwø,p[f], while if we chase it along the lefthand 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 onetoone 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 nfold 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 nfold 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 nfold 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 nfold 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##fflfflSPfrr
SP nE
The triangle on the righthand 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 CWcomplex 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 nfold 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 nfold covering maps associates to every nfold 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 nfold ramified covering maps (3.1).
Denote by Tn(h, k) the set of transfers for nfold ordinary covering maps,
and let E n ! B n be the universal principal nbundle. 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 nfold 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 nfold ramified
covering maps associated to it according to Corollary 3.7. Consider æ*[w] =
[w O æ] and let p : E  ! X be an nfold 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 nbundle 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_ naction 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 fflfflqqq
X ____'p____//SPnE_______SPng______//SPnH
where æ0 = . Since p : E ! X is an nfold 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 nfold 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 nfold 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 nfold ramified covering maps, for each n =
1, 2, 3, . .,.such that for each nramified 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+1fflfflfipnyysssss
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
nfold 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 qHnx n_n = ßn (and p O qSPnH = idSPnH).
Thus, combining (3.12) and the righthand 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+1fflfißnnttjjjjj
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] =
[effE ] 2 h(E) = [E, H] and [eff2] = [effX ] 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 wellpointed space, then the inclusion in : SP n1H ,! 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 wn1 = 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 wSPnH = wn. In particular, it
has the property that wH = idH. We have the following.
Lemma 5.5. Let H have the homotopy type of a connected CWcomplex.
If there is a map w : SP 1H  ! H such that wH = idH, then H has the
Q
homotopy type of a weak product en 0K(ßn(H), n), of EilenbergMac 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 DoldThom 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 CWcomplex.
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 EilenbergMac Lane spaces. 
Let now H be an abelian Hgroup 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( n1(a1, . .,.an1), 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 Hgroup 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 nfold 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 nfold 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 1fold 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 nfold 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 nfold 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##FSPfpflfflSPngmmmmmm_________________________
______________
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 nfold 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
 
 
fflffl0 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 nfold, resp.
n0fold, 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 nfold 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)ooooo  Jwn+wn0JJJ
ooo   JJJ
oo   %%
X NN   H:,:
NNN   tt
N0NNN   ttt
'(p,p )N''0fflffl fflfflwn+n0ttt
SP n+n (E t E0)________0____//SPn+n0H
SPn+n g
where wn + wn0 = O (wn x wn0). Hence, the righthandside 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 nfold 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 2fold 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 CWcomplex. By [12],
SPnH is also a CWcomplex and a subcomplex of SP n+1H. Therefore, we
have a (Hcoexact) 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 kdimensional vector bundles over X. By 3.7, given
an nfold 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
EilenbergMac 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 lefthand 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 multiindexes (permutations in the multiindexes 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 nfold 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 nfold 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) kdimensional Fvector
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 nfold 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 EilenbergMac 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 nfold 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)  ! S2k1, 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
1connected, 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
fflffliii
SP nBU (k)
30
seen as elements in He2(SP nBU (k)), where the top arrow on the righthand
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 1dimensional integral cohomology
We consider (H1(; Z), H1(; Z))transfers for nfold ramified covering maps
as well as for nfold 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 sZbundle_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 Zbundle __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 Zbundles; 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 nfold ramified covering maps are exactly the same as the
(H1(; Z), H1(; Z))transfers for nfold 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
nfold ramified covering map p : E  ! X and any element j 2 H1(E; Z).

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