Transfers for ramified coverings in homology
*
and cohomology
Marcelo A. Aguilar & Carlos Prieto1
Instituto de Matem'aticas, UNAM, 04510 M'exico, D.F., Mexico
Abstract
Making use of a modified version, due to McCord, of the Dold-
Thom construction of ordinary homology, we give a simple topological
definition of a transfer for ramified covering maps in homology with
arbitrary coefficients. The transfer is induced by a suitable map be-
tween topological groups. We also define a cohomology transfer which
is dual to the homology transfer. This duality allows us to show that
our homology transfer coincides with the one given by L. Smith. With
our definition of the homology transfer we can give simpler proofs of
the properties of the known transfer and of some new ones.
1 Introduction
In [16] L. Smith introduced a general class of finite ramified covering maps
and constructed for them a transfer in ordinary homology. Later on, in [6]
A. Dold gave an alternative construction and characterized ramified covering
maps as maps between orbit spaces of the action of a finite group and a
subgroup, and giving a modified definition of the transfer. Both definitions are
algebraic in nature. These transfers have the property that when composed
with the homomorphism induced by the projection of the ramified covering
______________________________
*2000 Math. Subj. Class.: Primary 55R12, 57M12; Secondary 55Q05, 55R35, 57M10
Keywords and phrases: Transfer, ramified covering maps, classifying spaces
E-mail addresses: marcelo@math.unam.mx, cprieto@math.unam.mx.
1This author was partially supported by PAPIIT grant No. IN110902.
1
map, they yield multiplication by the multiplicity of the covering in the
homology of the base space.
There are previous definitions of both the homology and the cohomology
transfers for maps between orbit spaces of certain actions of a finite group and
a subgroup, see Bredon [4] (and also tom Dieck [5]). These definitions depend
on the equivariant structure of the spaces involved. This is analogous to the
classical definition of the transfer for standard covering maps of Eckmann
[8], which is given at the level of chain complexes, thus it is rather algebraic
in nature.
In this paper we use a modified version, due to McCord [13], of the Dold-
Thom construction of ordinary homology to produce a topological transfer
for general ramified covering maps. Namely, we define a transfer that is a
map between some topological groups associated to the total and to the base
space of the ramified covering map. This codifies in a sense the fact that
a transfer can be seen as a multivalued map. We also define a cohomology
transfer using models of Eilenberg-Mac Lane spaces that have the structure
of topological abelian groups. We apply either transfer to give some results
about the homology or cohomology of orbit maps of the action of a group and
a subgroup of finite index. In contrast to previous transfers, whose definition,
and therefore also their computation, is more complicated than the definition
of the homomorphisms induced by the projections of the (ramified) covering
maps, our transfer definitions are somehow simpler than the latter, making
their computation easy (see (6.1), for instance).
In Section 2, for the benefit of the reader, we recall the construction of
McCord's topological groups and the nice main results of McCord's paper.
These are essential for our definition of the homology transfer and to prove its
properties. They also provide models of the Eilenberg-Mac Lane spaces which
are (weak) topological abelian groups. This fact will be used later to construct
the cohomology transfer. In Section 3 we recall the definition of a ramified
covering, and in Section 4 we define the homology transfer and prove some of
its properties. In Section 5 we define the cohomology transfer and prove some
of its properties. In Section 6 we use the transfers to prove some results about
group actions. For instance, we study the homology and the cohomology of
the orbit space of an arbitrary action of a compact Lie group in terms of the
orbit space of the action of the connected component of the identity element.
In Section 7 we show that the homology and the cohomology transfers are
dual with respect to the Kronecker product in homology and cohomology.
Finally, in Section 8 we prove that our homology transfer coincides with
2
Smith's transfer.
2 McCord's topological groups
In this section we recall briefly the spaces B(G, X) introduced by McCord.
We find it convenient to use F (X, G) as an alternative notation. Details can
be seen in [13] or [1, 6.3.20ff]. In this section we shall work in the category
of compactly generated weak Hausdorff spaces (see [13] or [12]) and therein
will be all spaces considered.
Definition 2.1. Let G be a topological abelian group and let X be a pointed
topological space with base point * 2 X. We define F (X, G) as the set of
all functions u : X - ! G such that u(*) = 0 and u(x) = 0 for all but a
finite number of elements x 2 X. If these elements are x1, . .,.xn and the
valuesPof u at each of them are g1, . .,.gn, it is sometimes convenient to write
u as ni=1gixi = g1x1 + . .+.gnxn. In particular, for any x 2 X, x 6= 0,
one may see gx as the element in F (X, G) whose value at x is g and whose
value elsewhere is 0 (g* = 0). Of course, these elements gx generate the
group F (X, G). Taking G = R to be a commutative ring with 1 and x 2 X,
then x 2 F (X, R) can be interpreted as the function whose value at x is
1 and whose value elsewhere is 0. This furnishes F (X, R) with a canonical
inclusion X ,! F (X, R). In this case, the elements x 2 F (X, R) generate
F (X, R) freely as an R-module.
Remark 2.2. In this paper we shall mainly restrict ourselves to abelian
groups G with the discrete topology, however we work in this section with
general topological abelian groups.
The set F (X, G) has a topology that turns it into a pointed space with
base point 0 2 F (X, G) the constant function with value 0, as we see below.
It is also an abelian group with the obvious addition. It is in fact a (weak)
topological abelian group. Consider the natural filtration
F0(X, G) F1(X, G) . . .F (X, G) ,
where Fn(X, G) consists of those functions u that are nonzero on at most n
points in X. If G = R, then one can define F1_2(X, R) = X F1(X, R), as ex-
plained above. The topology can then be defined as follows. For each n, take
the surjection k(G x X)n -! Fn(X, G) given by mapping (g1, x1, . .,.gn, xn)
3
P n
to i=1gixi. Here k(G x X)n is the product of n copies of G x X, furnished
with the compactly generated product topology, and Fn(X, G) is given the
corresponding quotient topology. Then provide F (X, G) with the weak topol-
ogy (of the union).
Given a pointed map ' : X - ! Y and a (continuous) homomorphism
ff : G -! H, one has a unique pointed map F (', ff) : F (X, G) -! F (Y, H)
given by
_ n !
X Xn
(2.3) F (', ff) gixi = ff(gi)'(xi) .
i=1 i=1
In other words, F (', ff)(u) is the function whose valuesPat y 2 Y are 0 unless
y = '(x) and u(x) 6= 0; in this case, F (', ff)(u)(y) = '(x)=yff(u(x)). This
definition turns F into a covariant bifunctor from the category Top *x Topab
of pairs consisting of a pointed topological space and a topological abelian
group to the category Topab of topological abelian groups.
We shall denote F (', 1G ) simply by '* and F (idX , ff) by ff*. The followi*
*ng
will be useful properties of the functor F :
(a) There is a natural isomorphism of topological abelian groups
F (X ^ Y, G) -! F (Y, F (X, G))
P P
given by mapping u = gi(xi^yi) to (gixi)yi, where gixi 2 F (X, G)
is as described above (see [13, 6.13]).
(b) There is a natural H-isomorphism (i.e., a homotopy equivalence that is
also a homomorphism) of H-groups
F (Y, G) -! F ( Y, G)
(if G is well pointed), where means the loop space and the (reduced)
suspension given by Y = S1 ^ Y . This H-isomorphism yields a group
isomorphism
oe : [X, F (Y, G)]* -! [X, F ( Y, G)]* ~=[ X, F ( Y, G)]* ,
where [-, -]* denotes pointed homotopy classes. We call this the sus-
pension isomorphism (see [13, 10.4]).
4
By (b), F (Sq, G) -'! F (Sq+1, G), and since F (S0, G) = G, we obtain
the following.
Theorem 2.4. ([13, 10.6]) The space F (Sq, G) is an Eilenberg-Mac Lane
space of type (G, q) that has the structure of an abelian group. ||
Note 2.5. This is not the first construction of Eilenberg-Mac Lane spaces
that yields topological abelian groups (for instance, Milnor [14] shows that
his construction of K(G, q) yields always a weak topological abelian group;
he shows that if K(G, q) is a countable CW-complex, then it is a topological
abelian group). However, McCord's construction is a very convenient one and
is easy to give.
Using property (b), we have a long exact sequence for the homotopy
groups ßq(F (A, G)), ßq(F (X, G)), and ßq(F (X [ CA, G) for a pair (X, A)
of the same homotopy type of a CW-pair . This and the previous theorem
prove the following.
Corollary 2.6. Let G be a discrete abelian group and let (X, A) be a pair of
spaces of the same homotopy type of a CW-pair. Then the homotopy groups
Hq(X, A; G) = ßq(F (X [ CA, G))
define an ordinary homology theory with coefficients in G. In particular, the
groups eHq(X; G) = ßq(F (X, G)) provide the reduced homology groups. More-
over, the groups of pointed homotopy classes
Hq(X, A; G) = [X [ CA, F (Sq, G)]*
define an ordinary cohomology theory with coefficients in G. In particular, the
groups eHq(X; G) = [X, F (Sq, G)]* provide the reduced cohomology groups. (If
A ,! X is a cofibration, one may of course replace X [ CA by X=A.) ||
Note that the groups of unpointed homotopy classes Hq(X; G) =
[X, F (Sq, G)] provide the unreduced cohomology groups.
Remark 2.7. If we assume that X is paracompact (instead of compactly
generated weak Hausdorff of the same homotopy type of a CW-complex),
then the groups [X, F (Sq, G)]* yield the ~Cech cohomology groups H~q(X; G)
(see [11]).
5
P m P m
Lemma 2.8. The map " : F (X, G) -! G given by i=1gixi 7! i=1gi is
well defined and continuous. In particular, " : F (S0, G) -! G is a homeo-
morphism.
Proof: This follows easily from the fact that the restriction "n : Fn(X, G) -!
G of " is continuous, since its composite with the identification (X xG)n -!
Fn(X, G) is obviously continuous. ||
Another useful property of the functor F is that one has a well-defined
continuous pairing
(2.9) F (X, G) x F (Y, H) -! F (X ^ Y, G H)
given by _ !
X X X
gixi, hjyj 7- ! (gi hj)(xi^ yj) ,
i j i,j
(see [13, 11.6]). If, in particular, G = H = R is a commutative ring with 1,
with m : R R -! R as the ring multiplication, then composing (2.9) with
m*, we obtain another pairing
(2.10) F (X, R) x F (Y, R) -! F (X ^ Y, R) .
Using (2.10), one obtains products in homology and cohomology. We shall
be interested in the following.
Proposition 2.11. One has cap-products
Hq(X; R) Hk(X; R) -`! Hk-q(X; R) ,
if X is 0-connected and q k, and
Hq(X; R) Hk(X; R) -`! Hq-k(X; R) ,
if k q. In particular, if k = q one has a Kronecker product
<-,->
(2.12) Hq(X; R) Hq(X; R) - ! R .
6
Proof: Taking smash-products and the pairing (2.9) we have
[X+ , F (Sq, R)]* x [Sk, F (X+ , R)]*_//[X+ ^ Sk, F (Sq, R) ^ F (X+ , R)]*
X X
X X X X |
~ XX X X X X,, fflffl||
[ kX+ , F ( qX+ , R)]* .
If q k, using oe-q of property (b), we desuspend q times. Composing ~ with
the homomorphism
[ k-qX+ , F (X+ , R)]* -! [Sk-q, F (X+ , R)]*
induced by the pointed inclusion S0 -! X+ that sends -1 to some point
x-1 in the path-connected space X, we obtain the homology `-product
`: [X+ , F (Sq, R)]* x [Sk, F (X+ , R)]* -! [Sk-q, F (X+ , R)]* .
On the other hand, if k q, using oe-k , we desuspend k times. And then,
composing ~ with the homomorphism
[X+ , F ( q-kX+ , R)]* -! [X+ , F (Sq-k, R)]*
induced by the obvious map X+ -! S0, we obtain the cohomology `-
product
`: [X+ , F (Sq, R)]* x [Sk, F (X+ , R)]* -! [X+ , F (Sq-k, R)]* .
In order to obtain the Kronecker product <-, ->, we take q = k and
consider the composite
[X+ , F (Sq, R)]*x[Sq, F (X+ , R)]* -`! [X+ , F (S0, R)]* -! [S0, F (S0, R)]* =*
* R ,
where the last arrow is induced by the pointed inclusion S0 -! X+ , and the
equality follows from the bijection " : F (S0, R) -! R given in Lemma 2.8.
||
3 Ramified coverings
We recall L. Smith's definition of a ramified covering map (see [16]). We shall
need the concept of nth symmetric product of Y defined by SP nY = Y n= n,
where n acts on the product of n copies of Y by permuting the coordinates.
We denote its elements by .
7
Definition 3.1. An n-fold ramified covering map is a continuous map p :
E - ! X together with a multiplicity function ~ : E - ! N such that the
following hold:
(i)The fibers p-1(x) are finite (discrete), x 2 X.
P
(ii)For each x 2 X, e2p-1(x)~(e) = n.
(iii)The map 'p : X -! SP nE given by
'p(x) = ,
~(e1) ~(em )
where p-1(x) = {e1, . .,.em }, is continuous.
Remark 3.2. Given an n-fold ramified covering map p : E - ! X with
multiplicity function ~, one can construct an n-fold ramified covering map
p+ : E+ -! X+ , where Y + = Y t {*} for any space Y and p+ extends p
by defining p+ (*) = * and the multiplicity function ~+ extends ~ by setting
~+ (*) = n. More generally, given a (closed) subspace A X, one can con-
struct an n-fold ramified covering map p0: E0 -! X=A, where E0 = E=p-1A,
p0 is the map between quotients and the multiplicity function ~0 coincides
with ~ off p-1A and is extended by setting ~0(*) = n, if * is the base point
onto which p-1A collapses. __
__ Another useful construction is the following. Let E = E t X and __p:
E - ! X be such that __p|E = p and __p|X = idX . Then __pis an (n + 1)-fold
ramified covering map with the obvious multiplicity function.
On the other hand, given a map F : Y -! X, one can construct the
induced n-fold ramified covering map F *(p) : F *(E) - ! Y by taking the
pullback F *(E) = {(y, e) 2 Y x E | F (y) = p(e)} and F *(p) = projY. The
induced multiplicity function F *(~) : F *(E) -! N is given by F *(~)(y, e) =
~(e). Call eF: F *(E) -! E the projection projE.
Examples 3.3. Typical examples of ramified covering maps are the follow-
ing:
1. Standard covering maps with finitely many leaves.
2. Orbit maps E= 0- ! E= for actions of a finite group on a space E
and 0 . They can be considered as [ : 0]-fold ramified covering
8
maps by taking ~(e 0) = [ e : 0e], where e and 0edenote the isotropy
subgroups of e 2 E for the action of and the restricted action of 0,
and [ : 0] and [ e : 0e] denote the corresponding indexes. In fact,
Dold [6] proves that all ramified covering maps are of this form for
= n and 0= n-1.
3. Branched covering maps on manifolds, namely open maps p : Md -!
Nd, where Md and Nd are orientable closed topological manifolds of
dimension d, p has finite fibers and its degree is n. Indeed, Berstein
and Edmonds [3] prove that p is of the form E= 0 -! E= , with
[ : 0] = n, so that, by 2., p is in fact an n-fold ramified covering
map. An interesting special case of this is given by Montesinos [15] and
Hilden [10], who show that for any closed orientable 3-manifold M3,
there is a branched covering map p : M3 -! S3 of degree 3.
4. It will be of particular interest to consider the following example. Let B
be a space and ßB : Bn x n __n-! SP nB, where __n= {1, 2, . .,.n} and
x n represents the twisted product, be given by ßB =
. Then ßB is an n-fold ramified covering map with mul-
tiplicity function ~B : Bn x n __n-! N given by ~B =
#{j | bj = bi} (see [16]).
4 The homology transfer
We shall define now the homolgy transfer. Our spaces in this section will be
compactly generated weak Hausdorff spaces.
Definition 4.1. Let p : E - ! X be an n-fold ramified covering map with
multiplicity function ~. Define the pretransfer
tp : F (X, G) -! F (E, G) by tp(u) = eu,
P n
where eu(e) = ~(e)u(p(e)). In other words, if u = i=1gixi 2 F (X, G), then
X
tp(u) = ~(e)gie .
p(e)=xi
i=1,...,n
Remark 4.2. The pretransfer tp : F (X, G) - ! F (E, G) is clearly a ho-
momorphism of topological groups and it is thus convenient to see what it
9
does to generators. Namely, if gx is the function in F (X, G) such that it is
zero everywhere, with the exception of x, where its value is g, then it is a
generator and the pretransfer satisfies
(
~(e)g if p(e) = x, i.e., if e 2 p-1(x)
tp(gx)(e) = ~(e)gx(p(e)) =
0 otherwise.
Hence, the only points where tp(gx) is nonzero are the elements of p-1(x) =
{e1, e2, . .,.er}, that is,
tp(gx)(e1) = ~(e1)g , tp(gx)(e2) = ~(e2)g , . .,.tp(gx)(er) = ~(er)g ,
and thus
tp(gx) = ~(e1)ge1 + ~(e2)ge2 + . .+.~(er)ger.
We shall prove below that tp is continuous. Hence, on homotopy groups,
the map tp induces the homolgy transfer
øp : eHq(X; G) -! eHq(E; G) .
We have the following.
Proposition 4.3. Let p : E -! X be an n-fold ramified covering map with
multiplicity function ~ : E -! N, where E and X are pointed spaces. Then
the pretransfer tp : F (X, G) -! F (E, G) is continuous.
Proof: Since F (X, G) has the topology of the union of the subspaces
. . .Fr(X, G) Fr+1(X, G) . . .F (X, G) ,
tp is continuous if and only if the restriction tp|Fr(X,G)is continuous for each
r 2 N. Denote by k(X x Y ) the product of X and Y with the compactly gen-
erated topology. Then we have a quotient map qr : k((G x X)r) i Fr(X, G)
for each r. Define ffi : G x X - ! Fn(X, G) by ffi(g, x) = tpq1(g, x) = tp(gx),
and ff : G x X -! (G x X)n= n by
ff(g, x) = [(g, e1), . .,.(g,,e1).,.(.g, em ), . .,.(g,]em,)
________-z_______" ________-z_______"
~(e1) ~(em )
where p-1(x) = {e1, . .,.em }. For each g 2 G, let ig : X - ! G x X be
given by ig(x) = (x, g), and let jg : En= n - ! (G x X)n= n be given by
10
jg[e1, . .,.en] = [(g, e1), . .,.(g, en)]. Then ff O ig = jg O 'p, where 'p : X*
* -!
En= n. Since jg and 'p are continuous and G is discrete, ff is continuous
and k(ff) : k(G x X) -! k((G x X)n= n) is also continuous. Since G x X is
compactly generated, k(GxX) = GxX. There is a natural homeomorphism
k((G x X)n= n) k((G x X)n)= n (indeed, it is straightforward to show
that the orbit space of the action of a finite group on a compactly generated
weak Hausdorff space is again a compactly generated weak Hausdorff space).
Therefore, the map k(ff) : G x X -! k((G x X)n)= n is continuous.
The quotient map qn factors through the quotient map q0n: k(G x X)n i
k((G x X)n)= n, yielding the following commutative diagram,
q0n n
k((G x X)n) ____////_k((G x X) )= n
lll
qn|| llllll
fflfflfflffl|jnuuuullll
Fn(X, G) ,
where æn is also a quotient map.
Now, ffi makes the following diagram commute,
k((G6x6X)n)= n
nnn
k(ff)nnnnn jn||
nnnnn fflfflfflffl|
G x X ___ffi_//_Fn(X, G) ,
therefore, ffi is continuous.
In order to show that tp|Fr(X,G)is continuous, consider the diagram
k(ffir)
k((G x X)r) _____//k(Fn(E, G) x . .x.Fn(E, G))
qr|| P|ri=1|
fflfflfflffl| fflffl|
Fr(X, G) ____t___________//F (E, G) ,
p|Fr(X,G)
P r
where i=1is the operation in F (E, G), which is a topological abelian group
in the compactly generated topology, and hence it is continuous. Since also
ffi is continuous, and qr is a quotient map, tp|Fr(X,G)is continuous. ||
Corollary 4.4. Let p : E - ! X be an n-fold ramified covering map with
multiplicity function ~ : E -! N, where E and X are pointed CW-complexes.
Then there is a homology transfer øp : eHq(X; G) -! eHq(E; G). ||
11
Remark 4.5. Besides the transfer øp defined above, for every integer k there
is another homology transfer kø given by (kø)p(,) = k . øp(,), , 2 Hq(X; G).
This transfer, in turn, is determined by the pretransfer (kt)p : F (X, G) -!
F (E, G) given by (kt)p(u) = k . tp(u), u 2 F (X, G).
Example 4.6. For the ramified covering map ßB : Bn x n __n-! SP nB of
3.3, the homology transfer is given as follows. We first compute
tiB : F (SP nB, G) -! F (Bn x n __n, G)
on the generators. Set
b = (b1,_._.,.b1_-z___", b2,_._.,.b2_-z___", . .,.br,_._.,.br_-z_*
*__") 2 Bn ,
i1 i2 ir
where i1 + i2 + . .i.r= n. Then
ß-1B** = {, . .,.**

} .
Therefore,
tiB(g) = ~**g**** + ~****g**** + . .+.
+ ~****g****
= i1g**** + i2g**** + . .+.irg****
= g **** + **** + . .<.b,+i1>
____________-z___________"
i1
+ **** + **** + . .+.****.+.
___________________-z__________________"
i2
+ **** + **** + . .+.****
____________-z___________"
ir
= g**** + . .+.**** + **** + . .<.b, i1 + i2> + . .+.
+ **** + ****
= g**** + g**** + . .+.g**** ,
hence
(4.7) tiB(g) = g) + . .+.g) .
12
P k
Thus, in general, if fi = i=1gi, then
(k,n)X
tiB(fi) = gi ,
(i,l)=(1,1)
since by varying l from 1 to n, the fiber elements over , namely
, are repeated ~B times.
Remark 4.8. Given an n-fold ramified covering map p : E - ! X with
multiplicity function ~ : E -! N, and a (closed) subspace A X, we
have the restricted ramified covering_map pA : EA - ! A, EA = pA , and the
quotient ramified covering map __p: E - ! X=A, as described in Remark 3.2.
The following diagram obviously commutes:
" __
EA Ø____//_E_____////_E
pA || p || |_p|
fflffl|Øfflffl|" fflffl|
A _____//_X___////_X=A .
Thus the diagram above yields
F (A, G)______//F (X, G)___//_F (X=A, G)
tA|| t|| |_t|
fflffl| fflffl| fflffl|_
F (EA , G)_____//F (E, G)____//_F (E , G) ,
_
where the horizontal arrows are_obvious and tA , t, and tare the corresponding
pretransfers. Therefore, using t, we have a relative homology transfer øp :
Hn(X, A; G) -! Hn(E, EA ; G), and by the commutativity of the diagram,
also this transfer maps the long exact sequences of (X, A) into the long
exact sequence of (E, EA ), provided that the inclusion A ,! X is a closed
cofibration (in general it is also true by constructing an adequate ramified
covering over X [ CA).
The following theorems establish the fundamental properties of the trans-
fer.
Theorem 4.9. The composite
p* O øp : eHn(X; G) -! eHn(X; G)
is multiplication by n.
13
The proof follows immediately from the following proposition.
Proposition 4.10. If p : E -! X is an n-fold ramified covering map, then
the composite
tp p*
F (X, G) -! F (E, G) -! F (X, G)
is multiplication by n.
P n iP n j
Proof: If u = i=1gixi 2 F (X, G), then p*tp(u) = p*tp i=1gixi =
P P n P P n
p(e)=xi, i=1,...,n~(e)gixi = i=1gixi p(e)=xi~(e) = n i=1gixi = n . u. *
* ||
The invariance under pullbacks is given by the following.
Theorem 4.11. Assume that F : X -! Y is continuous and that g : G -!
H is a homomorphism of discrete abelian groups. Then the following diagram
commutes:
fiF*(p)
Hq(Y ; G)_____//Hq(F *(E); G)
F*|| |eF*|
|fflffl fflffl|
Hq(X; G) __fip//_Hq(E; G) ,
where F *(E) -! Y is the n-fold ramified covering map induced by p : E -!
X over F .
As for the previous theorem, the proof follows immediately from the next
proposition.
Proposition 4.12. If p : E - ! X is an n-fold ramified covering map and
F : X -! Y is continuous, then the following diagram commutes.
tF*(p)
F (Y, G)_____//F (F *(E), G)
F*|| Fe*||
fflffl| fflffl|
F (X, G) __tp__//_F (E, G) .
14
P n
Proof: Let v = i=1giyi 2 F (Y, G). Then tF*(p)(v) 2 F (F *(E), G) is such
that
0 1
B X C
eF*tF*(p)(v) = eF*B F (~)(y, e)g (y, e)C
@ * i A
F*(p)(y,e)=yi
i=1,...,n
X
= ~(e)gieF(y, e)
F*(p)(y,e)=yi
i=1,...,n
X
= ~(e)gie
p(e)=F(yi)
i=1,...n
= tp (F*(v)).
||
One further property of the homology transfer that is useful is the fol-
lowing.
Proposition 4.13. Let f : B - ! C be continuous and consider the com-
mutative diagram
__ fnx n1_n n __
(4.14) Bn x n n __________//_C x n n
iB || |iC|
fflffl| fflffl|
SP nB _____SPnf____//SPnC .
Then the following diagram commutes:
(fnx n1_n)* n __
F (Bn x nO__n,OG)_________//_F (C OxOn n, G)
tßB || |tßC|
| |
F (SP nB, G)____(SPnf)___//F (SP nC, G) .
*
The proof is fairly routinary and follows easily using the description of
the transfers given in Example 4.6. ||
In 4.10 we computed the composite p*O tp. The opposite composite tpO p*
is also interesting. An immediate computation yields the following.
15
Proposition 4.15. Let p : E -! X by an n-fold ramified covering map with
multiplicity function ~. Then the composite
p* tp
F (E, G) -! F (X, G) -! F (E, G)
is given by X
tpp*(v)(e) = ~(e0)v(e0) ,
p(e0)=p(e)
for any v 2 F (E, G). ||
In the case of an action of a finite group on E and X = E= , we have
the following consequence.
P
Corollary 4.16. For v 2 F (E, G) one has tpp*(v)(e) = fl2 v(fle). There-
fore, the composite
p* tp
F (E= , G) -! F (E, G) -! F (E= , G)
P
is given by tpp*(v) = fl2 fl*(v).
Proof.Just observe that the element fle is repeated in the sum ~(e) = | e|
__
times. |__|
The two previous results yield the following in homology.
Theorem 4.17. Let p : E - ! X by an n-fold ramified covering map with
multiplicity function ~. Then the composite
p* fip
Hq(E; G) -! Hq(X; G) -! Hq(E; G)
is given by øpp*(y) = y0, where y0= [v0] 2 ßq(F (E, G)), and
X
v0(s)(e) = ~(e0)v(s)(e0)
p(e0)=p(e)
where y = [v] 2 ßq(F (E, G)) and s 2 Sq. ||
Corollary 4.18. For an action of a finite group on E and X = E= one
has that the composite
p* fip
Hq(E; G) -! Hq(E= ; G) -! Hq(E; G)
P
is given by øpp*(y) = fl2 fl*(y).
16
Remark 4.19. Considering an action of H on E and a subgroup K H,
one has different ramified covering maps as depicted in
E F
q0xxxx FFFqF
xx FFF
--xxx ""F
E= 0 _______0____//E= .
q
One may easily compute several combinations of the maps induced by these
covering maps and their transfers.
Another interesting property of the transfer is the relationship given by
computing the transfer of the composition of two ramified covering maps.
Before giving it we need the following.
Definition 4.20. Let p : Y -! X be an n-fold ramified covering map,
with multiplicity function ~ : Y -! N and let q : Z -! Y be an m-
fold ramified covering map, with multiplicity function : Z - ! N. Then
the composite p O q : Z - ! X is an mn-fold ramified covering map, with
multiplicity function , : Z -! N given by ,(z) = (z)~(q(z)). In order
to verify that this composite is indeed an mn-fold ramified covering map,
consider the wreath product n s m , defined as the semidirect product of
n and ( m )n, where n acts on ( m )n by permuting the n factors. We
have an action (Zm x . .x.Zm ) x n s m -! Zm x . .x.Zm given by
(i1, . .,.in) . (oe, ø1, . .,.øn) = (iff(1). ø1, . .,.iff(n). øn), where ii 2 Z*
*m . Then
we have the following diagram, where all maps are open
qx...xq m m
Zm x . .x.Zm __________//_Z = m x . .x.Z = m
i || i0||
fflffl| fflffl|
(Zm )n= n s m ` ` ` ` ` ` ` `//SPn(SP mZ) .
One may easily show that ß is compatible with ß0O (q x . .x.q). Therefore,
there is a homeomorphism Xmn = n s m SP n(SP mZ) and hence one has
a canonical quotient map æ : SP n(SP mZ) -! SP mnZ. Then one can easily
j mn
verify that 'pOq= æ O SP n('q) O 'p : X - ! SP n(SP mZ) -! SP Z. Thus
'pOqis continuous.
The homology transfer behaves well with respect to composite ramified
covering maps.
17
Theorem 4.21. The following hold:
tp tq
tpOq= tq O tp : F (X; G) -! F (Y ; G) -! F (Z; G) ;
fip fiq
øpOq= øq O øp : Hk(X; G) -! Hk(Y ; G) -! Hk(Z; G) .
Proof: As before, the second formula follows from the first. So, if u 2
F (X; G), v 2 F (Y ; G), w 2 F (Z; G), then v = tp(u) if v(y) = ~(y)u(p(y)),
and w = tq(v) if w(z) = (z)v(q(z)). Hence (tqtp(u))(z) = tq( (z)v(q(z)) =
(z)~(q(z))u(pq(z)) = ,(z)u((p O q)(z)) = tpOq(u)(z). ||
Corollary 4.22. Given an n-fold ramified covering map p : E - ! X with
multiplicity function ~ and an integer l, there is an ln-fold ramified covering
map pl: E -! X such that pl= p and ~l(e) = l~(e), e 2 E. Then tpl= ltp :
F (X; G) -! F (E; G) and øpl= løp : Hk(X; G) -! Hk(E; G).
Proof: Consider the l-fold ramified covering map q : E - ! E such that
q = idE and (e) = l for all e 2 E. Hence pl = p O q. Then apply Theorem
4.21. ||
Remark 4.23. The ln-fold covering map plobtained from p is a sort of spuri-
ous ramified covering map, since the multiplicity of p is artificially multipli*
*ed
by l. It is interesting to remark that the previous result shows that the trans-
fer of this new ramified covering map pl is just the corresponding multiple
of the transfer of the original ramified covering map p. Thus on this sort of
artificial ramified covering maps, the transfer remains essentially unchanged.
5 The cohomology transfer
In this section we define the chomology transfer and prove some of its prop-
erties.
Definition 5.1. Let p : E - ! X be an n-fold ramified covering map with
multiplicity function ~, where E and X are compactly generated weak Haus-
dorff spaces of the same homotopy type of CW-complexes. Define its coho-
mology transfer
øp : Hq(E; G) = [E, F (Sq, G)] -! [X, F (Sq, G)] = Hq(X; G)
18
P
by øp([eff]) = [ff], where ff(x) = p(e)=x~(e)eff(e), x 2 X. To see that the m*
*ap
ff is continuous and that its homotopy class depends only on the homotopy
class of eff, observe that ff is given by the composite
'p n SPneff n q q
ff : X -! SP E - ! SP F (S , G) -! F (S , G) ,
where the last map is given by the group structure on F (Sq, G). Using the
fact that X has the homotopy type of a CW-complex, similar arguments to
those used in the proof of 4.3 show that ff is continuous.
We might write øpninstead of øp when we wish to remark the multiplicity
n of the ramified covering map p.
Remark 5.2. We might assume that E and X are paracompact spaces in-
stead of compactly generated weak Hausdorff spaces of the same homotopy
type of a CW-complex. In this case, the same definition yields a transfer that
is a homomorphism between ~Cech cohomology groups
øp : ~Hq(E; G) -! ~Hq(X; G) ,
(see Remark 2.7), provided that G is an at most countable coefficient group.
Note 5.3. In order to define the cohomology transfer, the only property of
the Eilenberg-Mac Lane spaces given by F (Sq, G) required, is the fact that
they are (weak) topological abelian groups.
Similarly to the homology transfer, the cohomology transfer has the fol-
lowing fundamental properties.
Theorem 5.4. The composite
øpnO p* : Hk(X; G) -! Hk(X; G)
is multiplication by n.
Proof: If [ff] 2 [X, F (Sk, G)], then øpp*(ff) = øp(ff O p) : X - ! F (Sk, G),
P iP j
and øp(ffOp)(x) = p(e)=x~(e)ffp(e) = p(e)=x~(e) ff(x) = n.ff(x). Thus
øpp*([ff]) = n . [ff]. ||
19
Theorem 5.5. Let p : E - ! X be an n-fold ramified covering map and
assume that F : Y -! X is continuous. Then the following diagram com-
mutes: p
Hq(E; G) ___fi__//_Hq(X; G)
eF*|| |F*|
fflffl| fflffl|
Hq(F *(E); G) ___*_//Hq(Y ; G) ,
fiF (p)
where F *(p) : F *(E) -! Y is the n-fold ramified covering map induced by
p : E -! X over F .
Proof: Let eff: E -! F (Sq, G) represent an element in Hq(E; G). Then the
map X X
y 7- ! F *(~)(y, e)eff(y, e) = ~(e)eff(y, e)
F*(p)(y,e)=y p(e)=F(y)
*(p) * *p q
that represents øF eF(eff), clearly represents also F ø ([eff]) 2 H (Y ; G).
||
In 5.4 we computed the composite øp O p*. The opposite composite p*O øp
is also interesting. As it was the case for the homology transfer, an immediate
computation yields the following results for the cohomology transfer.
Proposition 5.6. Let p : E -! X be an n-fold ramified covering map with
multiplicity function ~. Then the composite
p q p* q
Hq(E; G) -fi!H (X; G) -! H (E; G)
is given as follows. Take ['] 2 Hq(E; G) = [E, F (Sq, G)], then p*øp['] is
represented by the map '0: E -! F (Sq, G) given by
X
'0(e) = ~(e0)e0.
p(e0)=p(e)
||
In the case of an action of a finite group on E and X = E= , we have
the following consequence.
Corollary 5.7. If , 2 Hq(E; G), then
X
p*øp(,) = fl*(,) 2 Hq(E; G), .
fl2
20
Proof: Just observe that in the sum the element fl*(,) is repeated ~(e) = | e|
times. ||
Generalizations and further properties of the cohomology transfer are
studied in [2].
6 Some applications of the transfers
First we start considering a standard n-fold covering map p : E - ! X.
In this case, the pretransfer (and thus also the transfer in homology) has a
particularly nice definition. Since the multiplicity function ~ : E - ! N is
constant ~(e) = 1, the transfer tp : F (X, G) -! F (E, G) is given by
(6.1) tp(u)(e) = u(p(e)) .
This fact has a nice consequence.
Theorem 6.2. Let be a finite group acting freely on a Hausdorff space
E. Then the orbit map p : E - ! E= is a standard covering map, and its
pretransfer induces an isomorphism
~=
tp : F (E= , G) -! F (E, G) ,
where the second term represents the fixed points under the induced -action
on F (E, G). Consequently, the pretransfer yields an isomorphism
~=
Hq(E= ; G) -! ßq(F (E, G) ) ,
for all q.
Proof.We assume that the projection p : E -! E= maps the base point to
the base point. The pretransfer tp is a monomorphism. Namely, if tp(u) = 0,
then, by (6.1), u(p(e)) = tp(u)(e) = 0 for all e 2 E. Since p is surjective,
u = 0.
On the other hand, obviously tp(u) 2 F (E, G) for all u 2 F (E= , G).
To see that it is an epimorphism, take any v 2 F (E, G) . Then v(e) = v(efl)
for all fl 2 , and thus v determines a well-defined element u 2 F (E= , G)
__
by u(e ) = v(e). Then clearly tp(u) = v. |__|
21
In what follows, we use the fundamental properties 4.9 and 4.18, and 5.4
and 5.7 of both the homology and the cohomology transfers to prove some
results about the homology and cohomology of orbit maps between orbit
spaces of the action of a topological group and a subgroup 0 of finite
index on a compactly generated weak Hausdorff space of the same homotopy
type of a CW-complex (and a corresponding result in ~Cech cohomology for
a paracompact space).
Before starting we need to recall Dold's definition of an n-fold ramified
covering map [6]. It is a finite-to-one map p : E - ! X together with a
continuous map _p : X -! SP nE such that
(i)for every e 2 E, e appears in the n-tuple _p(p(e)) = , and
(ii)SP n(p)_p(x) = 2 SP nX.
This definition is equivalent to Smith's (see 3.1), by setting 'p = _p and
defining ~(e) as the number of times that e is repeated in _p(p(e)).
We have the following interesting result.
Proposition 6.3. Let be a topological group acting on a space Y on the
right and let 0 be a subgroup of finite index n. Then the orbit map
p : Y= 0- ! Y= is an n-fold ramified covering map.
Proof: There is a commutative diagram
Y x __________//Y
| |
| |
fflffl| fflffl|
Y x ( = 0) ____//_Y= 0,
where the top map is the action and the vertical maps are the quotient maps.
Take the adjoint map of , j : Y - ! Map ( = 0, Y= 0). The function space
Map ( = 0, Y= 0) has a right -action given as follows. For f : = 0- ! Y= ,
take (f . fl)[fl1] = f(fl[fl1]) = f[flfl1]. The map j is then -equivariant and
thus induces a map
__j: Y= -! Map ( = 0, Y= 0)= .
On the other hand, if we identify = 0 with the set n_= {1, . .,.n}, then we
have a homeomorphism
Map ( = 0, Y= 0)= Map (n_, Y= 0)= n = SP n(Y= 0) .
22
Let _p : Y= -! SP n(Y= 0) be __jfollowed by the previous homeomorphism.
Then _p satisfies conditions (i) and (ii) and thus p is an n-fold ramified
covering map. ||
We apply the results 4.10 and 4.16 that we have for the pretransfer to
the n-fold ramified covering described above to obtain the following.
Proposition 6.4. Let Y be a space with an action of a topological group
and let 0 be a subgroup of finite index n. Assume that R is a ring
where the integer n is invertible. Then p* : F (Y= 0, R) -! F (Y= , R) is a
split (continuous) epimorphism. Moreover, if is finite and its order m is
invertible in R, then the kernel of p* is the complement in F (Y= 0, R) of the
invariant subgroup F (Y= 0, R) under the induced action of . Thus in this
case
F (Y= , R) ~=F (Y= 0, R) ;
in particular, if is finite and 0 is trivial, then m = n and
F (Y= , R) ~=F (Y, R) .
Proof: By 4.10 applied to the n-fold ramified covering p : Y= 0 -! Y= ,
p* O tp : F (Y= 0, R) - ! F (Y= 0, R) is multiplication by n, hence it is an
isomorphism, and consequently p* is a split epimorphism. Moreover, if is
finite of order m, by 4.16, we have that tpO p* : F (Y= 0, R) -! F (Y= 0, R)
is multiplication by m. So, if m is invertible in R, then p* : F (Y= 0, R) -!
F (Y= , R) is an isomorphism. ||
As an immediate consequence of the result above, or applying 4.9 and
4.18, we obtain the following two well-known results (cf. [16, 2.5], [4], [5]).
Theorem 6.5. Let Y be a space with an action of a topological group and
let 0 be a subgroup of finite index n. Assume that R is a ring where
the integer n is invertible. Then p* : Hq(Y= 0; R) -! Hq(Y= ; R) is a split
epimorphism. Moreover, if is finite and its order m is invertible in R, then
the kernel of p* is the complement of Hq(Y= 0; R) in Hq(Y= 0; R). Thus in
this case
Hq(Y= ; R) ~=Hq(Y= 0; R) ;
and in particular,
Hq(Y= ; R) ~=Hq(Y ; R) .
||
23
Similarly, 5.4 and 5.7 one has for cohomology the following.
Theorem 6.6. Let Y be a space with an action of a topological group and
let 0 be a subgroup of finite index n. Assume that R is a ring where
the integer n is invertible. Then p* : Hq(Y= ; R) -! Hq(Y= 0; R) is a split
monomorphism. Moreover, if is finite and its order m is invertible in R,
then the image of p* is Hq(X; R) . Thus in this case
Hq(Y= ; R) ~=Hq(Y= 0; R) ;
and in particular,
Hq(Y= ; R) ~=Hq(Y ; R) .
||
Remark 6.7. One may take a paracompact space Y with an action of a
topological group and obtain for ~Cech cohomology an analogous result,
namely p* : ~Hq(Y= ; R) -! ~Hq(Y= 0; R) is a split monomorphism, and
~Hq(Y= ; R) ~=H~q(Y= 0; R) .
A nice application of the previous ideas is the following generalization of
a well-known result of Grothendieck [9] (in the case Y = E ).
Theorem 6.8. Let be a compact Lie group and let 1 be the component
of 1 2 . Let R be a ring where n = [ , 1] is an invertible element. For an
action of on a topological space Y , one has
Hq(Y= ; R) ~=Hq(Y= 1; R) = 1,
Hq(Y= ; R) ~=Hq(Y= 1; R) = 1,
~Hq(Y= ; R) ~=H~q(Y= 1; R) = 1,
the last two according to what kind of a space Y is. ||
7 Duality between the homology and cohomology
transfers
In this section we compare the homology transfer with the cohomology trans-
fer.
24
Given an n-fold ramified cover p : E - ! X with multiplicity function
~ : E - ! N, we can extend it to the n-fold ramified covering map p+ :
E+ - ! X+ as explained in Remark 3.2. Consider the cohomology transfer
øp : Hq(E; G) = eHq(E+ ; G) -! eHq(X+ ; G) = Hq(X; G) ,
and consider also the homology transfer
øp : Hq(X; G) = eHq(X+ ; G) -! eHq(E+ ; G) = Hq(E; G)
as given in Definition 4.1.
Theorem 7.1. Let p : E -! X be an n-fold ramified covering map with mul-
tiplicity function ~ : E -! N and E path connected, and let øp : Hq(X; R) -!
Hq(E; R) and øp : Hq(E; R) -! Hq(X; R) be its homology and cohomolgy
transfers. If , 2 Hq(X; G) and e,2 Hq(E; G), then
<øp(,), e,>E = <,, øp(e,)>X 2 R ,
for the Kronecker products for E and X, respectively, and R a commutative
ring with 1 (see (2.12)).
Proof: We have to prove the commutativity of the following diagram:
[X+ , F (Sq, R)]*OxO[Sq, F (X+ , R)]*`//_[X+ , FM(S0, R)]*
| MMM
fipx1| MMM
| MMM
+ q q + M&&M
[E , F (S , R)]* x [S , F (X , R)]* q8R8q
qqq
1xfip|| qqqq
fflffl| qqq
[E+ , F (Sq, R)]* x [Sq, F (E+ ,_R)]*`//_[E+ , F (S0, R)]*
By the naturality of the construction of the pretransfers and the definition
of the `-product (see Proposition 2.11), it is fairly easy to check that this
commutativity follows from the commutativity of the following:
[X+ , F (X+O,OR)]*___//_[S0, F (X+M, R)]*
| MMM
fip| MMM
| MMM
+ + M&&M
[E , F (X , R)]* q8R8q
qqq
fip|| qqqq
fflffl| qqq
[E+ , F (E+ , R)]*___//[S0, F (E+ , R)]*
25
P m(e)
Let ffi : E+ -! F (X+ , R) be given by ffi(e) = i=1 ri(e)xi(e), e 2 E.
Chasing this element ffi along the top of the diagram, one easily verifies that
it maps to the element
X m(e)X
d = ~(e) ri(e) ,
p(e)=x-1 i=1
while chasing it along the bottom of the diagram, it maps to the element
m(e-1)X X m(e-1)X
d0= ri(e-1) ~(ei) = n ri(e-1) .
i=1 p(ei)=xi(e-1) i=1
P m(e)
Call æ(e) = i=1 ri(e). Since æ = Ö ffi, by 2.8 this defines a continuous map
æ : E -! R, but since E is path connected and R is discrete, æ is constant
with value rffi2 R. Hence
X
d = ~(e)æ(e) = n . rffiand d0= næ(e-1) = n . rffi.
p(e)=x-1
Thus d = d0 and the diagram commutes. ||
For simplicity, in what follows we omit the coefficient ring R in homology
and cohomology. For the Kronecker product <-, ->Y : Hq(Y ) Hq(Y ) -! R
there are induced homomorphisms Y : Hq(Y ) -! Hom (Hq(Y ), R) and Y :
Hq(Y ) - ! Hom (Hq(Y ), R) for every space Y , given by (y)(j) = Y
and (j)(y) = Y , y 2 Hq(Y ), j 2 Hq(Y ).
Corollary 7.2. The following diagrams commute
E X q
Hq(E) ______//Hom(Hq(E), R) Hq(X) _____//_Hom(H (X), R)
fip|| Hom|(fip,1)| fip|| |Hom(fip,1)|
fflffl| fflffl| fflffl| fflffl|
Hq(X) __X__//Hom(Hq(X), R) , Hq(E) __E_//_Hom(Hq(E), R) ,
the one on the right-hand side only if øp : Hq(E) -! Hq(X) is a homomor-
phism (which is rather seldom the case). ||
Remark 7.3. Under suitable conditions or are isomorphisms, in whose
case one of the transfers determines the other.
26
8 Comparison with Smith's transfer
In this section we show that the transfer defined in [16] coincides with ours
if we take Z-coefficients. To that end, we first recall his definition of the
transfer. It makes use of a result of Moore, that we state below. Recall that
Q 1
the weak product e n=1Xn of a family of pointed spaces is the colimit over n
of the directed system of spaces
X1 ,! X1 x X2 ,! X1 x X2 x X3 ,! . .,.
where the inclusions are given by letting the last coordinate be the base
point. Moore's result, as it appears in [18], is as follows.
Theorem 8.1. (Moore) A connected space X is weakly homotopy equivalent
Q
to the weak product en 1K(ßn(X), n) of Eilenberg-Mac Lane spaces if and
only if the Hurewicz homomorphism hn : ßn(X) - ! Hen(X; Z) is a split
monomorphism for all n 1. ||
Suppose that æn : eHn(X) = eHn(X; Z) -! ßn(X) is a left inverse of hn.
The Kronecker product defined in Section 2 determines an epimorphism
Hen(X; ßn(X)) -! Hom (Hen(X), ßn(X)) .
Let [,n] 2 eHn(X; ßn(X)) = [X, K(ßn(X), n)]* be some preimage of æn. Then
the family of pointed maps (,n) defines the weak homotopy equivalence of
the previous theorem.
Corollary 8.2. If X is a connected topological abelian monoid of the same
homotopy type of a CW-complex, then there is a homotopy equivalence X -!
eQ
n 1K(ßn(X), n).
Proof: Since X is a topological abelian monoid, there is a retraction r :
SP1 X -! X given by the retractions
rn : SP nX -! X , rn = x1 + x2 + . .+.xn .
Recall, on the other hand, that by the Dold-Thom theorem one has an iso-
morphism ßn(SP 1 X) ~=Hen(X), so that the inclusion i : X ,! SP 1X defines
the Hurewicz homomorphism (see [1]). Since r O i = idX, the homomorphism
æn = r* : Hen(X) = ßn(SP 1 X) - ! ßn(X) provides a left inverse of the
Hurewicz homomorphism hn. Hence, by Moore's theorem, we obtain the re-
sult. ||
27
Remark 8.3. Note that in the proof above, it is enough to assume that X is
a weak topological abelian monoid, i.e., that the product in X is continuous
on compact sets.
For any space E, the space SP 1E is a weak topological abelian monoid.
Thus we have the following.
Corollary 8.4. For a connected space E of the same homotopy type of
a CW-complex, there is a natural homotopy equivalence wE : SP 1 E - !
Q 1
K(He*(E)) = e n=1K(Hen(E), n). ||
The definition of Smith's transfer is as follows. Given an n-fold ramified
cover p : E - ! X with multiplicity function ~ : E - ! N, consider the
following composite:
'p n 1 '
bp: X -! SP E -! SP E -! K(He*(E)) .
This map defines a family of elements [bp] 2 He*(X; eH*(E)). On the other
hand, the Kronecker product determines a homomorphism
_ : eH*(X; eH*(E)) -! Hom (He*(X), eH*(E)) .
Smith's transfer is the image p] : eH*(X) -! eH*(E) of [bp] under the homo-
morphism _.
Theorem 8.5. Let p : E -! X be an n-fold ramified cover with multiplicity
function ~ : E - ! N. Then p] = øp : He*(X; Z) -! He*(E; Z), where øp is
the transfer in reduced homology.
Proof: Consider the following commutative diagram.
~=
[E, SPnE]* _____//[E, SP1 E]*___//_eH*(E, eH*(E))___//Hom(He*(E), eH*(E))
fip|| fip|| fip|| Hom|(fip,1)|
fflffl| fflffl|~ fflffl| fflffl|
[X, SPnE]* _____//[X, SP1 E]*=__//_eH*(X, eH*(E))__//_Hom(He*(X), eH*(E))
The two squares on the left-hand side, where øp represents the cohomology
transfer, commute obviously. The one on the right-hand side commutes by
Corollary 7.2. Take [i] 2 [E, SPnE]*, where i : E ,! SP nE is the canonical
inclusion. Chasing [i] down and then right on the bottom of the diagram,
28
we obtain p], while chasing it to the right on the top of the diagram and
then down, we obtain øp. This is true, because the image of [i] along the top
row of the diagram is the identity homomorphism 1 2 Hom (He*(E), eH*(E)).
This follows from the naturality of the Kronecker product, since by Corollary
8.2, we have an explicit description of the weak homotopy equivalence that
defines the isomorphism in the middle arrow. ||
References
[1]M. Aguilar, S. Gitler, C. Prieto, Algebraic Topology from a Ho-
motopical Viewpoint, Universitexts, Springer-Verlag, New York Berlin
Heidelberg 2002.
[2]M. Aguilar, C. Prieto, A classification of cohomology transfers for
ramified coverings, 2003 (to appear).
[3]I. Berstein, A. L. Edmonds, The degree and branch set of a branched
covering, Inventiones Math. 45 (1978), 213-220.
[4]G. E. Bredon, Introduction to Compact Transformation Groups, Aca-
demic Press, New York and London 1972.
[5]T. tom Dieck, Transformation Groups, de Gruyter, Berlin, New York
1987
[6]A. Dold, Ramified coverings, orbit projections and symmetric powers,
Math. Proc. Camb. Phil. Soc. 99 (1986), 65-72.
[7]A. Dold, R. Thom, Quasifaserungen und unendliche symmetrische
Produkte, Annals of Math. 67 (1958), 239-281.
[8]B. Eckmann, On complexes with operators, Proc. Nat. Acad. Sci. 39
(1953), 35-42
[9]A. Grothendieck, `Sur quelques points d'algebre homologique'
T^ohoku Math. J. 9 (1957), 119-221
[10]H. M. Hilden, Every closed orientable 3-manifold is a 3-fold branched
covering space of S3, Bull. Amer. Math. Soc. 80 (1974), 1243-1244.
29
[11]P. J. Huber, Homotopical cohomology and ~Cech cohomology, Math.
Annalen 144 (1961), 73-76.
[12]J. P. May, A Concise Course in Algebraic Topology, Chicago Lecture
Notes in Mathematics, The University of Chicago Press, Chicago and
London 1999.
[13]M. C. McCord, Classifying spaces and infinite symmetric products,
Trans. Amer. Math. Soc. 146 (1969), 273-298.
[14]J. W. Milnor, The geometric realization of a semi-simplicial complex,
Annals of Math. 65 (1957), 357-362
[15]J. M. Montesinos, Three-manifolds as 3-fold branched covers of S3,
Quart. J. Math. Oxford Ser. (2), 27 (1976), 85-94
[16]L. Smith, Transfer and ramified coverings, Math. Proc. Camb. Phil.
Soc. 93 (1983), 485-493.
[17]G. W. Whitehead, Elements of Homotopy Theory, Graduate Texts
in Math., Springer-Verlag, New York Heidelberg Berlin 1978.
December 3, 2003
30
**