Two lower bounds for the
relative LusternikSchnirelmann category
PierreMarie MOYAUX
October 2, 1998
Abstract
We prove that oep+1cat(X)+1 cat(X; X xSp) and that e(X; X x
Sp) = e(X) + 1, where oep+1cat is the oecategory of Vandembroucq
and e is the Toomer invariant. The proof is based on an extension to
a relative setting of Milnor's construction of the classifying space of a
topological group.
______________________________
0PierreMarie MOYAUX; Universite de Lille 1; U.F.R. de Mathematiques & U.R.A*
* 751
au CNRS; 59655 Villeneuve D'Ascq, France.
email address : moyaux@gat.univlille1.fr
0AMS subject classification : 55M30, 57R70.
0Key Words : LusternikSchnirelmann category, critical points estimates.
1
1 Introduction
The LusternikSchnirelmann category is a numerical invariant of the homo
topy type of a space, first defined in [13] . If X is a topological space, cat(*
*X)
is the smallest k such that X can be covered by (k+1) open sets contractible
in X. The L.S. category has been extended to many other contexts. In par
ticular, Fadell [7] defined the category of a pair (X,A) of spaces, cat(X,A).
This is the smallest k such that X can be covered by k+1 open sets (Ui)0ik
such that A is a retract of U0 and the other open sets are contractible in X.
We show that oep+1cat(X) + 1 cat(X; X x Sp) (*), where oeicat(X) is
the stable L.S. category defined by Vandembroucq [22]. For i 2 N, oeicat(X)
is the least integer k such that the ifold suspension of the kth Ganea fi
bration ipk(X) : iGk(X) ! iX has a homotopy section. (In particular,
oe0cat(X) = cat(X)).
The initial motivation for the study of the L.S. category is given by the
fact that the number of critical points of a smooth function defined on a
manifold M is bounded below by cat(M) + 1. The relative invariant has also
applications to critical points estimates. To any smooth function h defined
on a riemannian manifold X, (possibly with boundary), we associate the flow
induced by  5 h. If A is the part of the boundary of X where this flow
points outward, cat(X; A) is a lower bound for the number of critical points
of h [4].
One reason for the interest in the relationship between oep+1cat(X) and
cat(X; X x Sp) comes from the following particular case. If M is a closed
manifold, we may consider the functions defined on M x Dn having as exit
set (for the associated flow) M x Dnt1 x St @(M x Dn). The number of
critical points of such functions is bounded below by cat(M; M xSt) [4]. This
class of functions defined on M x Rn contains those quadratic at infinity. In
turn, these are important in symplectic geometry. Estimating their number
of critical points in terms of invariants of M is one of the central problems
in the stable Morse and L.S. Theory of Eliashberg and Gromov [6]. Define
a suspension of a flow to be the product of this flow with a gradient one
associated to a quadratic form on Rn. Many of the properties of this new
flow are related to stable homotopical properties of M (cf [3] ). Moreover,
the suspension of a gradient flow on M may be viewed as the gradient flow
2
of some function quadratic at infinity defined on M x Rn. This suggests
that the number of critical points of a function quadratic at infinity may be
estimated by a more stable invariant associated to M.
This is what we prove here with (*), since oep+1cat is precisely such an
invariant, and cat(M; M x St) is a lower bound for the number of critical
points of the functions quadratic at infinity.
A natural question is whether we can replace in (*) oep+1cat by another
bigger homotopy invariant inv . The Arnold conjecture seemed to sug
gest that we might have taken inv = cat, i.e. that we had cat(X) + 1
cat(X; X x Sp) . Together with cat(X; X x Sp) cat(X x Sp) cat(X) + 1,
proved in [4], this inequality would have implied cat(X x Sp) = cat(X) + 1.
This statement is known as the Ganea conjecture, and has been recently dis
proven by Iwase [12] . However, for a symplectic, closed manifold M of sym
plectic form !, satisfying !ss2(M)= 0, one has cat(M; M x Sp) = cat(M) + 1.
[18]
The following section is devoted to recalls concerning the Borel construc
tion and the relative category. In the 3  rd section, adapting [17], we give a
relative construction of the classifying space of a topological group (depend
ing on both the group G and a Gspace F ). We also prove that this con
struction can be obtained by Ganea's fibercofiber construction. In the last
section, we obtain explicit links between relative and absolute Milnor/Ganea
spaces in the particular case of a trivial Gaction on F . Exploiting these
links, we obtain (*) and prove that e(X) + 1 = e(X; X x Sp), where e is the
(relative) Toomer invariant.
Aknowledgements : This article is part of the author's PhD disserta
tion. The author is supported by a scholarship from the French Department
of Higher Education. The author wishes to thank his advisor, Octav Cornea,
for his guidance and his patience. This work was partially realised at the
University of Rochester. The author wishes to thank the U. of R. Depart
ment of Mathematics for their hospitality and kindness, in particular Fred
Cohen and John Harper for useful discussions and comments. The author
also thanks Maxence Cuvilliez for a useful suggestion.
3
2 Preliminaries
In this paper, we work in the category of compactly generated spaces, written
here CGspaces (see [20], [23]). In particular, all cartesian products are
retopologized (and thus group actions, trivialization of bundles, etc... are
defined with the retopologized product). Note also that for the definition of
fiber bundles and principal Gbundles, we use coverings of the base space by
regular open sets.
2.1 The Borel construction
In this subsection, we fix a few standard results concerning quotients of
spaces by group actions. Here, the spaces are endowed with the action of a
CG topological group G. If A is a Gspace, A=G represents the orbit space
of A and is given the quotient topology. (Thus, if A=G is Hausdorff, it is
also CG). The quotient map A ! A=G is denoted qA . If a 2 A, then qA (a)
is usually denoted by [a].
Lemma 2.1 Let ABCD be a pushout square with equivariant arrows, the
action on D being induced by the action on B and C. Then, the square ob
tained from ABCD by taking the quotient spaces and maps (w.r.t. the action
of G) is also a pushout square.
Proof : Consider the following commutative diagram :
A=G ___B=G
 A
  A
 
? ? A
C=G ___D=G A
H A
H H A
OHHE A
HHj AAU
E
As ABCD is a pushout square, the maps O qB and OE O qC induce a
map i : D ! E. As both maps are constant on each orbit, i is also con
stant on each orbit, and thus can be factored through D/G. Then, the map
i: D=G ! E makes the above diagram commute. Thus, the quotient square
satisfies the universal property of pushout squares. o
4
In general, the quotient of a space F by a group action does not give rise to
qF
a principal Gbundle G ! F  ! F=G. However, we have the following (cf
[1]):
Lemma 2.2 (Borel's construction) Let : G ! E ! B be a principal
GBundle. Then, for any space F endowed with a G action, we have the
following diagram, where :
i) both rows are fibre bundles of fiber F,
ii) both columns are principal Gbundles, (the action of G on E x F is diag
onal),
iii) the bottom right square is a pullback square,
=
G ____________G
 
 
 
 
 
 
 
? prE ?
F __________Ex F __________E
  
  
  
=  qExF  qE 
  
  
  
? ? ss ?
F ________(Ex F )=G ________B
(Here, ss is the map induced by prE ).
Proof : The proof is standard. For example, we have for i):
As is a principal Gbundle, B is covered by trivializing open sets (Ui)i2I.
If U 2 (Ui)i2I, we have homeomorphisms :
 OE : q1E(U) ! U x G defined by OE(e) = ([e]; '(e)) , with '(e:g) = '(e):g;
 = OE1 : U x G ! q1E(U) .
Moreover, by definition, the map E xG ! E xE, defined by (e; g) ! (e; eg),
is a homeomorphism on its image E xB E. Let denote the inverse of this
map composed with the projection on G. Thus, if e and e0 are points in the
same orbit of E, (e; e0) is the unique element of G satisfying e:(e; e0) = e0.
U is also a trivializing open set for ss. The chart maps are given by :
 : ss1(U) ! U x F , defined by ([e; f]) = ([e]; f:'(e)1);
 : U x F ! ss1(U) defined by ([e]; f) = [e; f:'(e)].
5
These applications are well defined and inverse homeomorphisms. Thus,
(Ui)i2Iis a covering of B trivializing for ss .
If b = [ffl] is any point in B, then ss1(b) = {[e; f] s.t. [e] = [ffl]} is hom*
*eomor
phic to F:
 ss1([ffl]) ! F is defined by [e; f] ! f:(e; ffl);
 F ! ss1([ffl]) is defined by f ! [ffl; f].
Thus, i) is proved. o
Two particular cases of this construction will be of interest :
Corollary 2.3 i) if F is contractible and B admits a numerable trivializing
covering for qE , then and ss*() have the same homotopy type.
ii) (G x F )=G ! F defined by [g; f] ! f:g1 is a homeomorphism, and the
map (G x F )=G ! F=G, induced by the projection on F, can be identified
with the quotient map qF : F ! F=G.
Proof : i) Using the chart maps given in the proof of Lemma 2.2.i, as F is
contractible, we see that ssss1(U): ss1(U) ! U is a fiber homotopy equiva
lence over U. As (Ui)i2Iis numerable, the result follows from Thm 3.3 of [5].
(See also Thm 1.5 of [15]).
ii) The proof of the first part is immediate. For the second part, just re
mark that (G x F )=G ! F=G is the composition of this homeomorphism
with qF . o
2.2 Relative L.S. category
In this subsection, we assume the spaces and maps to be well pointed. Here,
we give some basic facts about the relative L.S. category and fix a few nota
tions.
First, let us recall a tool used to study the L.S. category: the fibercofiber
construction (FCconstruction), due to Ganea [9] .
Given a map f : A ! X, we turn it into a fibration ^f: ^A! X, and we take
the fiber F of ^f. Since (X; *) is an N.D.R. pair, (A^; F ) is also an N.D.R. p*
*air,
i.e. F ,! ^Ais a cofibration. We take its cofiber ^A=F . Then, we iterate these
operations with "p1, the map induced between A^=F and X. (i.e. we turn it
into a fibration p1 : G1(X; A) ! X, take the fiber F1(X; A), and form the
quotient G1(X; A)=F1(X; A), with an induced map going to X ...).
6
Remark : i) Ganea proved in [10] that Fk+1(X; A) ' Fk(X; A) * X.
ii) The FCconstruction, when applied to CGspaces, provides CGspaces.
iii) When A is a point, Ganea's construction is identified with Milnor's con
struction [17] of the classifying space of X (cf [8]).
pk(X;A)
Notations : For any k, Fk(X; A) ,! Gk(X; A) ! X is a fibration.
We define qk(X; A) : A ! Gk(X; A) by qk(X; A) = tk1 O ::: O t0. As in [4],
we identify A and its image by qk(X; A). Thus, pk(X; A) may be viewed as
a map of pairs, pk(X; A) : (Gk(X; A); A) ! (X; A).
The maps obtained by applying the FCconstruction to f are denoted by
pk(X; A) and qk(X; A). If A is a point, we just denote them by pk(X) and
qk(X). When there is no ambiguity about the spaces we consider, we simply
denote them by pk and qk.
We can summarize the FCconstruction in the following diagram :
F _______F * X _______:::_______Fk(X;A) _______:::
    
    
    
i0(X;A) i1(X;A)  ik(X;A) 
    
    
? t0 ? t1 ? ? tk ?
A ' ^A_ A^=F ' G1(X; A) ___:::p______Gk(X;A) _____p_:::
H H pp pppp
H H @ pp ppppp
H H @ p1(X;A)pppk(X;A) ppppp
p0(X;A) H H @ ppp ppppp
HH@ pp pppp
Hj@@R? sps
X
In the following, let (X,A) be a N.D.R. pair, and let f denote the map
A ,! X.
Ganea in [9], gave an equivalent definition of cat(X). He proved that the
map pk(X) admits a homotopy section s iff cat(X) k. In [4], this type of
definition is extended to a relative setting.
Definition 2.1 The category of (X,A) is the smallest k such that pk(X; A)
7
admits a homotopy section s verifying s O f ' qk(X; A) .
Remark : We may define cat(X,A) for an arbitrary map f : A ! X, by
using Def 2.1 . We obtain the same result as if turning f into a cofibration
"f: A ! X" and computing cat(X"; A) with the definition given in the intro
duction.
Example : Apply the FCconstruction to the Hopf fibration j : S3 ! S2.
We obtain :
S1 _______S1* S2 _S1 * S2 * S2
  
  
  
i0 i1 i2
  
  
? t0 ? t1 ?
S3 _______S3_ S2 ____G2(S2;S3)


 p1
j  p
 2

? ss
S2
We already know (cf [4]) that cat(S2; S3) cat(S2) + 1 = 2. It is easy to
see that the restriction of p1 to S2 is a homotopy equivalence. In particular,
p1 admits a homotopy section. However, q1 = t0 is the injection of S3 in the
wedge, and thus can not be factored through S2. So, cat(S2; S3) 6= 1. Thus,
cat(S2; S3) = 2.
3 The Relative Milnor construction
Let G be a CG topological group, and F a CGspace endowed with a right
Gaction. In the following, all Gbundles are principal Gbundles.
Milnor, in [17], has constructed for each n an nuniversal bundle G !
En(G) ! Bn(G). BG, the classifying space of G, is the direct limit of
(Bn(G))n2N . Thus, we have B0(G) = * B1(G) ::: Bn(G) ::: BG.
Here, we give a filtration of BG by spaces depending on both G and F .
8
More precisely, we construct, for each n 2 N, spaces Bn(G; F ), satisfying
B0(G; F ) = (EG x F )=G B1(G; F ) ::: Bn(G; F ) ::: whose direct
limit B(G; F ) has the same homotopy type as BG.
We recall that if A and B are two spaces, A * B, the (non reduced) join
of A and B, is the space obtained from A x I x B by taking the quotient
by the relations (a; 0; b) = (a0; 0; b) and (a; 1; b) = (a; 1; b0) for any a; a*
*02 A,
b; b02 B. A*B is given the quotient topology. Thus, if A and B are CG, A*B
is CG. Note that A*B is homeomorphic to the pushout of AxB ! CAxB
and of A x B ! A x CB.
In [17], En(G) is defined as the join of (n + 1) copies of G, with the strong
topology. Note that if G is a CGgroup, Milnor's construction can also be
done with En(G) having the usual quotient topology. ( The keypoint in
the choice of the topology of the join is to have a continuous action of G on
En(G), and continuous "coordinates" maps for the join.)
We define the spaces En(G; F ) by induction :
 E0(G; F ) = F x EG;
 En+1(G; F ) = En(G; F ) * G = (F x EG) * En(G);
 E(G; F ) = (F x EG) * EG.
These new spaces are CG, and thus endowed with the diagonal Gaction.
We recall that EG (and thus E(G; F )) is contractible (cf [19]).
Notations :
 Bn(G; F ) is the quotient space En(G; F )=G, with the quotient topology.
 OEn(G; F ) : En(G; F ) ! Bn(G; F ) is the projection.
 rn(G; F ) : Bn(G; F ) ! Bn+1(G; F ) is the quotient of the equivariant map
En(G; F ) ,! En+1(G; F ).
As in the absolute case, we have :
OEn(G;F)
Lemma 3.1 For any n, G ! En(G; F ) ! Bn(G; F ) is a Gbundle.
Proof : For n = 0, the result follows from the Borel construction.
For n 1, remark that we can construct Bn(G; F ) by applying the Borel
construction to F * En(G) endowed with the diagonal action. In fact, we
have the following diagram, where all the vertical maps are permutations of
9
coordinates A x B x C ! A x C x B.
F x EG x En(G)___CF x EG x En(G)
 pp
 @ pp @
 @@R ppp @@R
 __pp_
 F x EG x EG pp (F x EG) * En(G)
  pp 
  pp 
  pp 
?  ? 
ppp ae
F x En(G) x EG  CF x En(G) x EG n
 p 
@  pp 
 pp 
@@R ? R ?
F x EG x EG _____(F* En(G)) x EG
Since the front left and back arrows are equivariant homeomorphisms, they
induce between the pushouts an equivariant homeomorphism aen. The quo
tient of aen is also a homeomorphism. Thus, we obtain :
G ______________En+1(G; F)__________Bn+1(G;F)
  
  
  
 ae~ ~
 n= =
  
  
? ? ?
G ____________(F* En(G)) x EG____(F_*_En(G))_x_EGG
As the lower row is a Gbundle, OEn+1(G; F ) : G ! En+1(G; F ) ! Bn+1(G; F )
is also a Gbundle. o
From now on, we identify these two Gbundles. Clearly, this argument re
mains valid if we replace OEn(G) by any principal Gbundle. We express the
above property by saying that the Borel construction commutes with the join.
Notations :
 0(G; F ) : B0(G; F ) ! BG is the quotient of F x EG ! EG.
n+1(G; F ) : Bn+1(G; F ) ! BG is the quotient of (F * En(G)) x EG ! EG.
It follows from the Borel construction that for any n, n+1(G; F ) is a fiber
10
bundle with fiber F * En(G).
 OEn(G) denotes the projection map of the bundle En(G) ! Bn(G) obtained
at the n  th stage of the usual Milnor construction; OE(G) denotes the map
EG ! BG.
When there is no ambiguity about G and F, the maps OEn(G; F ), n(G; F )
and rn(G; F ) will be denoted by OEn, n, rn.
Remark : i) Take F = G, and the multiplication as a right action. It
follows from corollary 2.3.i (applied with = OEn+1(G)) that our construc
tion is equivalent to the one in [17], i.e. we may identify OEn(G; G) and OEn(G*
*).
Thus, we identify (EG x En(G))=G with Bn(G), and (EG x EG)=G with
BG.
ii) Note that, also by using corollary 2.3.i, we obtain that : B(G; F ) ! BG,
quotient of (F * EG) x EG ! EG, is a homotopy equivalence.
iii) With the notations of Lemma 2.2, suppose G, E, B and F are CG (and
thus Hausdorff). As (E xF )=G is the total space of a fibre bundle with Haus
dorff base space and fiber, it is also Hausdorff. Then, as qExF is a proclusio*
*n,
(E x F )=G is CG. In particular, Bn(G; F ) is CG, for any n.
Proposition 3.2 The relative Milnor construction applied to a space F en
dowed with an action of a topological group G can be identified with the FC
construction applied to 0(G; F ) : B0(G; F ) ! BG.
For any n 0, we may identify :
 pn(BG; B0(G; F )) with n(G; F ) : Bn(G; F ) ! BG,
 tn(BG; B0(G; F )) with rn(G; F ) : Bn(G; F ) ! Bn+1(G; F ),
 in(BG; B0(G; F )) with OEn(G; F ) : En(G; F ) ! Bn(G; F ).
Proof : We first show how to obtain n+1(G; F ) from n(G; F ) by induction.
Construction of the cube:
The upper square is the construction of En+2(G; F ) as join of En+1(G; F )
and G. (Here, we take En+1(G; F ) = (F * En(G)) x EG, cf Lemma 3.1).
Using Lemma 2.1, we take the quotient of this square and obtain Bn+2(G; F )
expressed as a pushout. We identify the spaces and maps on the left and
lower faces by using corollary 2.3.ii. (The left vertical maps are of the form
(e; g) ! e:g1). Moreover, since C(F * En(G)) x EG is contractible, we see
that Bn+2(G; F ) is the homotopy cofiber of OEn+1.
11
En+1(G; F) x_G______En+1(G;pF)
@ pp@p
 @ pp @
 @@R ppp @@R
 C(F * E (G)) x EG pppx_GE (G; F)pppppppppppppppprEG
 n p n+2 EG
  pp  
  pp  
  pp  
  pp  
?  ?  
E (G; F)pppppppppBpp(G; F)  
n+1  n+1 p  
 pp  
@  p  
@  pp  
@  pp  
@R ? R ? n+2(G;F) ?
C(F * En(G)) x EG_____Bn+2(G; F)ppppppppppppBGppp
Each vertex of the upper face is a product of EG with another space, and
the projection En+2(G; F ) ! EG is induced by the other projections on EG.
We take the quotient of these maps, and obtain that n+2(G; F ) is induced
by n+1(G; F ) and the map o : C(F * En(G)) x EG ! BG, where o is the
quotient of C(F * En(G)) x EG x G ! C(F * En(G)) x EG ! EG.
By corollary 2.3.ii, we identify o with the composite C(F * En(G)) x EG !
(C(F * En(G)) x EG)=G ! BG, which is equal to C(F * En(G)) x EG !
EG ! BG .
Now, we apply the FCconstruction to n+1(G; F ), ( in the upper face of
the following cube), and compare the resulting map "n+2(G; F ) with the
map n+2(G; F ).
Construction of the cube :
Both upper and lower faces are pushout squares.
 The map i is the injection of F * En(G) in (F * En(G)) x EG.
 The back right map r is the injection of C(F *En(G)) in C(F *En(G))xEG
 The front left map is the identity on Bn+1(G; F ).
As these maps are homotopy equivalences, they induce a homotopy equiva
lence h between the pushouts.
12
F * En(G)________C(F *pEn(G))
@ pp@p
 @ pp @
 @@R ppp @@R
i B (G; F)____ppp__" ppppppppppppppp"(n+2G;F)
 n+1 p Bn+2(G; F) BG
  pp  
  pp  
  pp  
  pp  
?  ?  
E (G; F)ppppppC(Fp*pE(G)) x EGh =
n+1  n p  
 pp  
@  p  
@  pp  
@  pp  
@R ? R ? n+2(G;F) ?
Bn+1(G; F)_________Bn+2(G; F)ppppppppppppBGppp
" n+2(G; F ) : B"n+2(G; F ) ! BG is induced by n+1(G; F ) and the trivial
map on the cone.
n+2(G; F ) : Bn+2(G; F ) ! BG is induced by n+1(G; F ) and o.
As the composition o O r is the constant map, it follows from the univer
sal property of pushout squares that "n+2(G; F ) = n+2(G; F ) O h. Thus,
we obtain the stated result by functoriality of the fiber cofiber construction.*
* o
Example : Take F = Z=4Z and G = Z=2Z . The action Z=2Z x Z=4Z !
Z=4Z is defined by (a; b) ! 2a+b. As the quotient of F by the Gaction gives
rise to a Gbundle, we may take En+1(Z=2Z; Z=4Z) = Z=4Z * En(Z=2Z).
Let X be a Z=2Zspace, with an action . Taking the (unreduced) join of X
with Z=2Z is exactly the same as taking the (unreduced) suspension of X.
If we consider X as the union over their base of two cones C0X and C1X
(C0X = C1X = X x I=X x {0}), then tx (1  t)i 2 X * Z=2Z corresponds
to the point [x; t] in CiX. With this identification, X is a Z=2Zspace.
Thus, in our case, we may identify En+1(Z=2Z; Z=4Z) with the pushout
of two copies of Sn ! Sn+1, each sphere being endowed with the antipo
dal action. Thus, it follows from Lemma 2.1 that Bn+1(Z=2Z; Z=4Z) is the
pushout of two copies of RP n! RP n+1 .
13
4 Lower bounds for the relative category
4.1 The relative Toomer invariant
Here, we use the notations of Section 2.2 . The relative category (as well as
the absolute one) is generally hard to compute, and thus, its approximations
are valuable. In particular, Toomer [21] defined an invariant e related to the
homological properties of the FCconstruction : e(X) is the smallest number
k such that the map pk(X) is surjective in homology.
Clearly, we have e(X) oeicat(X) cat(X), for all i 2 N.
We also define a Toomer type invariant in the relative case:
Definition 4.1 The Toomer invariant of (X,A), denoted by e(X,A), is the
smallest k such that pk(X; A) induces a surjection in homology :
pk(X; A)* : H*(Gk(X; A); A) ! H*(X; A).
Equivalently, we may require (pk(X; A)=A)* : H*(Gk(X; A)=A) ! H*(X=A)
to be surjective. As in the absolute case, we obtain :
Proposition 4.1 : For any N.D.R. pair (X,A), e(X; A) cat(X; A).
Proof : Suppose that cat(X; A) = k.
Then, there is a map s : X ! Gk(X; A) such that pkOs ' idX and sOf ' qk.
We want to show that pk , considered as a map of pairs, also admits a section.
As pk is a fibration, we may assume that pk O s = idX . Moreover, it is proved
in [11] that the homotopy H between s O f and qk may be chosen vertical.
Then, since f : A ,! X is a cofibration, we can construct a homotopy section
"sof pk such that "sA= qk.
Because H is vertical, "s: (X; A) ! (Gk(X; A); A) is a homotopy section of
the map pk considered as a map of pairs.
Thus, (pk)* : H*(Gk(X; A); A) ! H*(X; A) is surjective. o
We have just bounded below the relative category by another relative invari
ant. In the following, among other things, we express, in the case (X; A) =
(X; X x Sp), this (relative) lower bound in terms of an absolute invariant.
4.2 Relative Milnor construction and relative category
In the following, G acts trivially on F, and the action of G on a product
or join of spaces is the diagonal action. (Recall that in this case, we have
14
B0(G; F ) = BG x F ).
The relative Milnor spaces associated to the trivial action have particular
properties.
Theorem 4.2 Assume that G acts trivially on F then :
i)For any n, there is a homotopy commutative diagram:
F * Bn(G) _ F __________________Bn+1(G;'F )=B0(G; F )
 
 
F* n(G)_idF  n+1(G;F)=B0(G;F)
 
 
? ' ?
F * BG _ F _______________________BG=B0(G;F )
ii) For any n, there is up to homotopy, a cofibration sequence:
fln
Bn+1(G; F ) ____________F* Bn(G) __________F * BG _ BG
iii)For any n, there is a homotopy commutative diagram:
Bn+1(G; F )
 Q Q
'  Q Q n+1(G;F)
 QQ
? QQs
F * (BG=Bn(G)) _ BG ____BGprBG
15
Proof: Call L the left diagram, and R the right diagram.
F x EG x En(G)___CF x EG x En(G) F x EG x En(G)___CF x EG x En(G)
 pp  pp
 @ pp @  @ pp @
 @@R ppp @@R  @@R ppp @@R
 __pp_  ___pp_
 F x EG x EG pp (F x EG) * En(G) F x EG x EG pp (F x EG) * En(G)
  pp    pp 
  pp    pp 
  pp    pp 
?  ?  ?  ? 
EG_x_En(G)_ppp EG_x_En(G)_ EG_x_En(G)_ppp EG_x_En(G)_
F x G  CF x G  F x G  CF x G 
  p p    p p 
 @  pp pp   @  pp pp 
  pp pp    pp pp 
 @@R ? pp R ?  @@R ? pp R ?
 EG_x_EG___pp_(F_x_EG)_*_En(G)_ EG_x_EG__pp_ EG_x_En(G)_
 F x G pp G  F x C G pp F * G
  pp    pp 
  pp    pp 
  pp    pp 
?  ?  ?  ? 
pppppppppppp  pppppppppppp 
F x BG  CF x BG  F x BG  CF x BG 
 p   p 
@  pp  @  pp 
 pp   pp 
@@R ? R ? @@R ? R ?
F x BG__________CF x BG F x C(BG)__________F * BG
Construction of L :
The upper face of L is a pushout and the middle face of L is obtained
by taking its quotient by the action of G. Using Lemma 2.1, we see that the
middle face of L is a pushout. The lower face of L is obtained by taking the
direct limit (following n), of the spaces and maps of the middle face. (Recall
that we identify EGxEn(G)_Gwith Bn(G) and EGxEG__Gwith BG).
Construction of R :
The upper and middle faces of R are pushout squares.
In the upper cube, the vertical back arrows are the projections on the quo
tient spaces, the vertical front left map is the projection on the quotient
space composed with the obvious injection, and the vertical front right map
is induced by the other. The lower face is obtained by taking the limit of the
spaces and maps of the middle face.
16
Now, we construct maps between L and R :
We take the identity map on each space of the upper face and on the two
spaces on the back of the middle face, and the obvious injection F x EGxEG__G!
F xC(EGxEG__G). These maps induce a map fln : (FxEG)*En(G)_G! F * EGxEn(G)_G.
The maps between the lower faces are the limits of the maps between the
middle squares.
i) We consider L and take the cofibers of the front and back horizontal maps.
To identify the map obtained on the right, i.e. coming from the front face,
note that the quotient of the map (F x EG) x EG ! (F x EG) * En(G) cor
responds to the map B0(G; F ) = F x BG ! Bn+1(G; F ) , and that the limit
of this map is 0.
ii)
F x EG_x_En(G)_G_CF x EG_x_En(G)_G
pp
@ pp@
 @@R ppp @@R
 pp
 F x EG_x_EG____p_(F_x_EG)_*_En(G)_
 G pp G
 pp
  pp 
  pp 
?  ? 
F x EG_x_En(G)_ppCFpxpEG_x_En(G)_fln
G  G 
 pp 
@  p 
@  pp 
@R ? R ?
EG x EG EG x En(G)
F x C _______G____F * __________G
We construct the above cube by taking the middle faces of L and R and the
maps already defined between them. The square constituted by the cofibers
of the vertical maps is a pushout square. As the vertical back maps are
identities and the cofiber of A x B ! A x CB is homotopic to A * B _ B,
this proves ii).
iii) By Lemma 13, 15 of [14], since the upper, lower and back faces of the
above cube are pushout squares, so is the front face. We construct the cube
having this square as upper face, and the direct limit of this square as lower
17
face.
F x EG_x_EG_G___(F_x_EG)_*_En(G)_G
pp
@ pp@
 @@R ppfl@@Rnp
 pp
=  F x C EG_x_EG___p_F * EG_x_En(G)_
 G pp G
 pp
  pp 
  pp 
?  ? 
F x BG ppppppppppppBGpppp F* n(G)
 
 pp 
@  p 
@  i pp 
@R ? R ?
F x C BG __________F* BG
As the right square is a pushout square (also proved by using [14]), the *
*map
h induced between the cofiber of fln and the cofiber of i : BG ! F * BG is
a homotopy equivalence.
We obtain the following diagram, where the back rows are cofibration se
quences.
Bn+1(G; F)___________Ff*lBn(G)n_____________Cof(fln)pppppppppppBn+1(G;pF)*
*ppp
   pp pp
  @  pp pp
  @ l h  p p
  '  pp' pp
  @  pp pp
  @@R  p pp
  pppppppppppppp pp
n+1(G;F) F* n(G) cof(i)  cof(i)=(F * BnG) p n+1*
*(G;F)
    pp pp
    pp pp
    pp pp
    pp pp
    pp pp
? ?  ? pp ?
BG ________________F* BG ______________cof(i)ppppppppppppppppBGppppp*
*p
 pp pp
@  pp pp
@  pp pp
 = pp pp=
@  pppp
@@R ? ?
cof(i)ppppppppppppppppBGpppp
We take the cofibers (represented in dotted arrows) of the horizontal and
18
diagonal maps departing from F * Bn(G) and from F * BG. The map
F * BG ! cof(i) may be identified with F * BG ,! F * BG _ BG.
F* n(G)
Then, as l is the composite F * Bn(G)  ! F * BG ! cof(i), we identify
cof(i)=(F * BnG) ! BG with the projection F * cof( n(G)) _ BG !
BG. o
Remark : The idea of constructing pn+1(X; A) as the join of pn(X) and
f, which is used in L, was suggested to the author by Maxence Cuvilliez.
Corollary 4.3 : For any space X of the homotopy type of a CW complex,
and any p 1 we have :
i) e(X; X x Sp) = e(X) + 1,
ii) oep+1cat(X) + 1 cat(X; X x Sp):
Proof : X is homotopy equivalent to BG, for some CW group G. Then,
by fonctoriality, we may identify the FCconstruction applied to prBG :
BGxF ! BG and to prX : X xF ! X. In particular, X (resp. (X; X xF ))
and BG (resp. (BG; BG x F )) have the same homotopy invariants. More
over, if we consider F endowed with the trivial G action, 0(G; F ) = prBG .
It follows from prop 3.2 that Ganea's construction associated to prBG and
Milnor's construction associated to (G,F) can be identified. Moreover, as
usual, we identify the FC construction applied to the Path Loop fibration of
BG with the usual Milnor's construction applied to G.
Apply Theorem 4.2.i with F = Sp.
i) The homology of Sp * Bn(G) (resp Sp * BG) is just the homology of Bn(G)
(resp BG) with the degrees shifted. Clearly, the surjectivity in homology of
n+1(G; Sp)=B0(G; Sp) is equivalent to the surjectivity in homology of n(G).
ii) cat(BG; BG x Sp) = n + 1 implies the existence of a section to the map
n+1(G; Sp)=B0(G; Sp) (cf proof of Prop 4.1). In turn, this implies the ex
istence of a section to Sp * n(G) _ idSp, and thus to Sp * n(G). Thus,
oep+1cat(X) n. o
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