A GENERALIZATION OF THE TRIAD THEOREM OF
BLAKERS-MASSEY
WOJCIECH CHACHOLSKI
1. Introduction
The purpose of this paper is to find a geometric reason for the triad theorem*
* of
Blakers-Massey (see [1] and [7]). We are asking the following question:
How far a homotopy push-out square is from being homotopy
pull-back?
In particular, for a cofibration map A ! X, we want to investigate the differen*
*ce
between A and the homotopy fiber of the cofiber map X ! X=A.
The initial data for our investigation can be organized by having a commutati*
*ve
diagram:
A P---! E P
? P P Pq? P P Pid
?? P?P?Pq PP Pq
y y Y ---! E
? ?
X P---! B P ?? ??
P P Pid P PyPid y
PP Pq PP Pq
X ---! B
where the following are respectively a homotopy push-out square and a homotopy
pull-back square:
A ---! E Y - --! E
?? ? ? ?
?y ??y ??y ??y
X ---! B X - --! B
The map q : A ! Y measures how far the above homotopy push-out diagram is
from being homotopy pull-back. Our aim is to understand this map. In this paper*
* we
will be able to give some "approximation" for the homotopy fiber and the homoto*
*py
cofiber of q : A ! Y . We will approximate using cellular inequalities - a *
*notion
that was introduced by E. Dror Farjoun (see [6]).
Let A be a connected space. By C(A) we denote the smallest class of connected
spaces that contains A and is closed under weak equivalences, taking homotopy p*
*ush-
outs, arbitrary wedges and telescopes. One can think about the class C(A) as ge*
*n-
erated by A using certain simple operations: taking homotopy push-outs, arbitra*
*ry
1
2 WOJCIECH CHACHOLSKI
wedges and telescopes. We say that a space X is built by A if X belongs to C(A).
If X is built by A, we write X >> A. We refer to the relation ">>" as a strong
cellular inequality.
The following theorem is the main result of this paper. It gives a cellular e*
*stimation
for the homotopy fiber and the homotopy cofiber of q : A ! Y (see corollar*
*y 7.4
and theorem 7.5):
Theorem. Let A, E and X be connected. If F ib(A ! X) and F ib(A ! E) are
connected, then:
o F ib(q : A ! Y ) >> F ib(A ! X) ^ F ib(A ! E)
o Cof(q : A ! Y ) >> F ib(A ! X) ? F ib(A ! E)
where W ^ V and W ? V are respectively the wedge and the join of W and V .
Let A ! X be a cofibration and F be the homotopy fiber of the quotient map
X ! X=A. Under an additional assumption, out of the above theorem, we can get a
cellular estimation for the homotopy cofiber Cof(q : A ! F ) entirely in t*
*erms
of A and X=A (see corollary 6.9):
Theorem. Let A be connected. If A ! X is a cofibration, for which X=A is weakly
equivalent to the suspension of a connected space, then:
Cof(q : A ! F ) >> A ^ (X=A)
The above cellular approximations are derived from an inequality that general*
*izes
Serre's theorem [9, theorem 6.1] (see theorem 5.2):
Theorem. Let the following be a homotopy pull-back square:
Y ---! E
?? ?
?y ??y
X ---! B
If B is connected, then:
F ib(E=Y ! B=X) >> F ib(E ! B) ? F ib(X ! B)
where E=Y and B=X are the homotopy cofibers, respectively of Y ! E and X ! B.
As an application of the proven inequalities one can get the triad theorem of
Blakers-Massey (see corollary 7.7). In addition one can also get statements as *
*follows
(see corollary 7.6):
Theorem. Let B be connected. Let the following be a homotopy push-out square:
A ---! E
?? ?
?y ??y
X ---! B
A GENERALIZATION OF THE TRIAD THEOREM OF BLAKERS-MASSEY 3
Let Y = holim(X ! B E) and q : A ! Y be the natural map. Let p and
s be distinct prime numbers. If the reduced integral homology of F ib(A ! E)
and F ib(A ! X) are respectively p and s torsion, then q : A ! Y is a homology
isomorphism.
2. Notation
This paper is written simplicialy. The only place though where we use in a cr*
*ucial
way the nature of spaces, we are working with, is the proof of the main theorem
(see 5.1). All the statements remain valid in the category of topological space*
*s having
the homotopy type of a CW-complex.
The homotopy cofiber of a map A ! X is denoted either by X=A, if it is clear
which map we are considering, or by Cof(A ! X), if we want to point out the map.
Lets choose a basepoint in X. The homotopy fiber of a map A ! X at the chosen
basepoint is denoted by F ib(A ! X). If X is connected, then the homotopy type *
*of
F ib(A ! X) does not depend on the choice of a basepoint in X.
For a space X, by X we denote the unreduced suspension of X. If we choose a
basepoint in X, by X we denote the homotopy fiber of the basepoint map ? ! X.
Let X and Y be pointed spaces. The subspace X x {y0} [ {x0} x Y of X x Y
is denoted by X _ Y and is called the wedge of X and Y . The quotient space
X x Y=(X _ Y ) is denoted by X ^ Y and is called the smash of X and Y . For any
choice of basepoints in X and S1, X is weakly equivalent to X ^ S1 (in the case*
* of
topological spaces, we have to assume that the chosen basepoint in A is a cofib*
*ration).
Let K be a simplicial set. By the same symbol K we denote the category of
simplices of K (see [3, definition 3.1]). Objects of this category are simplic*
*es of K
and morphisms are generated by arrows oe ! dioe and oe ! sioe, where dioe is th*
*e i-th
face of a simplex oe and sioe is the i-th degeneracy of oe. Functors over the c*
*ategory
associated with a simplicial set K are called diagrams over this simplicial set*
*. If for
every simplex oe 2 K, F (oe ! sioe) = id and F (oe ! dioe) is a weak equivalenc*
*e,
then F is called a good diagram (see [3, definitionI3.8]). Having such a diag*
*ram
F : K ! Spaces , we can form its homotopy colimit F . The homotopy colimit
I K
F is a simplicial set, whose set of n-simplices is given by:
K
iI j a
F = F (oe)n
K n oe2Kn
n0
The references regarding the homotopy colimit construction of diagrams over s*
*im-
plicial sets are [3] and [4]. In this paper the only place, where we are going*
* to use
this notion, is the proof of the main theorem (see theorem 5.1).
4 WOJCIECH CHACHOLSKI
3. Closed classes
In this section we state the definition and give some examples and basic prop*
*erties
of closed classes. We will also discus the notion of strong cellular inequaliti*
*es. The
references are [2], [3] and [6]. In this section we state a language in which a*
* general-
ization of the triad theorem is going to be expressed. In the rest of the paper*
* we are
going to use, in an essential way, cellular techniques that are sketched in thi*
*s section.
These techniques were originally introduced by E. Dror Farjoun (see [6]).
Definition 3.1. A non empty class C of connected spaces, that does not contain*
* the
empty space, is a closed class if the following conditions are satisfied:
o Let X and Y be weakly equivalent. If X 2 C, then Y 2 C.
o Let (Xi)i2IbeWa family of spaces. If Xi2 C, then for any choice of basep*
*oints
in Xi, i2IXi2 C.
o Let (X1 X2 ! X3) be a push-out diagram. If Xi2 C, then:
hocolim(X1 X2 ! X3) 2 C
o Let ( X0 ! X1 ! X2 ! . .).be a diagram. If Xi2 C, then:
hocolim(X0 ! X1 ! X2 ! . .).2 C
Example 3.2. Let A be a connected space. By C(A) we denote the smallest closed
class that contains A. The elements of this class are called A-cellular spaces *
*(see [2]
and [5] for detailed discussion of this class).
A good way of thinking about the class of A-cellular spaces is that it is gen*
*erated
by A using certain simple operations: taking arbitrary wedges, homotopy push-ou*
*ts
and telescopes.
Example 3.3. Let n 0. Let the class Cn consists of those connected spaces X,
such that for any choice of a basepoint in X, fHi(X) = ?, for i n, where fHi(X*
*) is
the reduced integral homology of X.
Definition 3.4. (E.Dror Farjoun [6]). Let A be a connected space and X be a sp*
*ace.
We write X >> A, if X 2 C(A). We call the relation ">>" a strong cellular
inequality.
Remark. An expression X >> A is defined only for a connected space A. Whenever
an expressions X >> A appears in this paper, it is understood that A is connect*
*ed.
The following theorem is one of the main tools to study closed classes. It i*
*s a
generalization of E. Dror Farjoun's theorem (see [3, theorem 9.1] and [6]).
Theorem 3.5. Let K be a connected simplicial set. Let : E ! B be a natu*
*ral
transformation betweenidiagrams E : K !jSpaces and B : K ! Spaces . If for ev*
*ery
simplex oe 2 K, F ib oe : E(oe)! B(oe) belongs to a closed class C, then so d*
*oes
i I I j
F ib : E ! B .
K K
A GENERALIZATION OF THE TRIAD THEOREM OF BLAKERS-MASSEY 5
A very useful particular cases of this theorem are listed in the following pr*
*oposition:
Proposition 3.6.
o Let the following be a map between homotopy push-out diagrams of connect*
*ed
spaces: i j
E? ' hocolim E1 - E2 -! E3
?? ?? ?? ??
y i ?y ?y ?y j
B ' hocolim B1 - B2 -! B3
If for i = 1; 2; 3, F ib(Ei! Bi) >> A, then F ib(E ! B) >> A.
o Let the following be a homotopy push-out diagram of connected spaces:
f
A ---! E
?? ?
?y ??y
g
X ---! B
If F ib(f : A ! E ) is connected, then F ib(g : X ! B ) >> F ib(f : A ! *
*E )
o Let A ! X be a cofibration map between connected spaces. Looking at the
following homotopy push-out square:
A ---! ?
?? ?
?y ??y
g
X ---! X=A
we get: F ib(X ! X=A) >> A.
The following proposition lists some basic cellular inequalities. These inequ*
*alities
were originally discovered by E. Dror Farjoun.
Proposition 3.7.
o ? >> A (see [5, section 2.3]).
o Let A ! X be a map over a connected space X. If this map has a right
homotopy inverse (A is a retract of X), then A >> X (see [5, lemma 6.1]).
o Let A ! X be a map. If X is connected, then X=A >> F ib(A ! X)
(see [6]).
o Let A be simply-connected. If X >> A, then X >> A (see [2, corollary
10.4]).
o Let A be connected. X >> A if and only if X is simply connected and
X >> A (see [2, theorem 10.8]).
o A >> A.
o If A is simply-connected, then A >> A.
As a corollary we get:
6 WOJCIECH CHACHOLSKI
Corollary 3.8. Let A ! X be a map over a connected space X. If F ib(A ! X) 2
Cn, then X=A 2 Cn+1 (see example 3.3).
Proposition 3.9. Let D be connected and B be a pointed space. The following c*
*lass
is closed:
C = {Y | Y is connected and for any choice of a basepoint in YY,^ B >> D}
Proof. It is obvious that C is closed under weak equivalence. We have to prove *
*that
C is closed under taking arbitrary wedges, homotopy push-outs and telescopes.
Let (Y1 Y2 ! Y3) be a push-out diagram of spaces that belong to C. Let
Y = hocolim(Y1 Y2 ! Y3). Since Y ^ B is weakly equivalent to the homotopy
push-out hocolim(Y1 ^ B Y2 ^ B ! Y3 ^ B), we get: Y ^ B >> D and Y 2 C.
The proofs, for the class C being closed under taking arbitrary wedges and te*
*le-
scopes, go in the same way. __|_ |
Corollary 3.10. Let B be a pointed spaces. If X >> A, then X ^ B >> A ^ B.
Proof. According to proposition 3.9, the following class is closed:
C = {Y | Y is connected and for any choice of a basepoint in YY,^ B >> A ^ B}
Obviously A 2 C, therefore the smallest closed class C(A), that contains A, is *
*in-
cluded in C. __|_ |
Since taking the suspension of a space is weakly equivalent with smashing this
space with S1, we get:
Corollary 3.11. If X >> A, then X >> A.
4. Join
In this section we present definition and basic properties of the join constr*
*uction.
Definition 4.1. The join X ? Y of two spaces X and Y is defined as follows:
p1 p2
X ? Y = hocolim(X X x Y ! Y )
Proposition 4.2. For any choice of basepoints in X and Y , X ? Y ' (X ^ Y ).
A GENERALIZATION OF THE TRIAD THEOREM OF BLAKERS-MASSEY 7
Proof. Let us consider the following commutative diagram:
? --- X --id-! X --- ? - --! ? ?
x? x x x x x
?? ???id ???pr1 ??? ??? ???
? --- X --i1-!X x Y --i2- Y - --! ? X ^ Y
?? ? ? ? ? ?
?y ??y ??ypr2 ??yid ??y ??y
? --- ? ---! Y --id- Y - --! ? ?
? --- ? ---! X ? Y --- ? - --! ?
Applying the homotopy colimit functor first to the columns of this diagram and *
*then
to the obtained row, we get: X ? Y . Reversing this procedure and applying the
homotopy colimit functor first to the rows of the diagram and then to the obtai*
*ned
column, we get: (X ^ Y ). This proves: X ? Y ' (X ^ Y ). __|_ |
Corollary 4.3. If fHi(X) = ? for i n and fHi(Y ) = ? for i m, then fHi(X ?Y *
*) = ?
for i n + m + 2.
i *
* j
Proposition 4.4. If X is connected, then closed classes C(X ?Y ) and C (X ?Y*
* )
are the same.
Proof. The space X is connected, therefore X ? Y is simply connected (it is the
suspension of a connected space) and proposition 3.7 implies: X ? Y >> (X ? Y ).
Since X ? Y ' (X ^ Y ) and X ^ Y is connected, using again proposition 3.7, we
obtain (X ? Y ) >> X ^ Y . Suspending this inequality gives:
(X ? Y ) >> (X ^ Y ) ' X ? Y
__
|_ |
Proposition 4.5. If X >> A, then X ? Y >> A ? Y .
Proof. Let us choose a basepoint in Y . Corollary 3.10 implies: X ^ Y >> A ^ Y *
*. By
suspending this inequality, we get: X ? Y ' (X ^ Y ) >> (A ^ Y ) ' A ? Y . __|*
*_ |
As a straightforward consequence of the proposition, we get:
Corollary 4.6. If X >> A and Y >> B, then X ? Y >> A ? B.
8 WOJCIECH CHACHOLSKI
5. Main theorem
In this section we are going to prove a strong cellular inequality which will*
* be our
main technical tool to prove a generalization of the triad theorem. This inequ*
*ality
can be thought of as a generalization of Serre's theorem (see [9, theorem 6.1]).
Theorem 5.1. Let B be a connected space. Let us consider the following commu*
*ta-
tive diagram:
H ______________! F Z
| @ ?? ??
| @ ?y ?y
| @
|| @R l
| Y ---! E ---! E=Y
| ?? ?? ??
| ? ?p ?
|# y f y y
g
G - --! X ---! B ---! B=X
where:
o F ! E is the homotopy fiber of p : E ! B ,
o G ! X is the homotopy fiber of g : X ! B ,
o H ! Y is the homotopy fiber of the composition gOf : Y ! B ,
o E ! E=Y is the homotopy cofiber of l : Y ! E,
o B ! B=X is the homotopy cofiber of g : X ! B ,
o maps H ! F , H ! G and E=Y ! B=X are induced by l, f, p and g,
o Z ! E=Y is the homotopy fiber of E=Y ! B=X.
If hocolim(G H ! F ) is connected, then:
Z >> hocolim(G H ! F )
Remark. Notice that since B is connected, the homotopy fibers G ! X, F ! E,
H ! Y and Z ! E=Y are unique up to homotopy.
Proof. By changing the diagram in a homotopy meaningful way we can arrange so
that the maps g : X ! B , p : E ! B and gOf : Y ! B are fibrations.
According to [3, example 3.12], we can construct good diagrams: G : K ! Space*
*s ,
F : K ! Spaces , H : K ! Spaces together with natural transformations : H ! G
and : H ! F , such that:
o for every simplex oe 2 K, G(oe), H(oe) and F (oe) are weakly equivalent *
*respec-
tively to the homotopy fibers G, H and F ,
A GENERALIZATION OF THE TRIAD THEOREM OF BLAKERS-MASSEY 9
o the following commutative squares are weakly equivalent:
I I
Y - -l-! E KH ---! K F
?? ? ?? ??
?yf ??yp ?y ?y
I
g
X - --! B G ---! K
K
It follows that the map E=Y ! B=X can be represented, up to homotopy, as a
homotopy push-out:
i I I j
E=Y ' hocolim ? - H -! F
?? ? K K?
?y ??y # ?y
i I j
B=X ' hocolim ? - G -! K
K
Without changing its homotopy colimit, we can modify the above diagram as follo*
*ws:
i I id I I I j
E=Y ' hocolim ? - G -! G - H - ! F
?? ? K K K K
?y ??y # id # id # #
i I id I id I I j
B=X ' hocolim ? - G -! G - G - ! ?
K K K K
Once again, without changing its homotopy colimit, we can modify the last diagr*
*am
further, obtaining the following weak equivalence of maps:
i I I j
E=Y ' hocolim ? - G -! hocolim(G H ! F )
?? ? K K
?y ??y # id #
i I I id j
B=X ' hocolim ? G ! hocolim(G G ! ?)
K K
i id j
Since for every simplex oe 2 K, hocolim G(oe) G(oe) ! ? is contractible, *
*the
map: I I
hocolim(G H ! F ) ! hocolim(G id G ! ?)
K I K
is weakly equivalent to hocolim(G H ! F ) ! K. According to theorem 3.5,
K iI j
we get that the homotopy fiber F ib hocolim(G H ! F ) ! K is built by
K
hocolim(G H ! F ), where G, H and F are the homotopy fibers respectively of
g, gOf and p. By applying proposition 3.6, we get:
F ib(E=Y ! B=X) >> hocolim(G H ! F )
__
|_ |
10 WOJCIECH CHACHOLSKI
The following theorem is a very useful particular case of theorem 5.1.
Theorem 5.2. Let the following be a homotopy pull-back square:
Y --l-! E
?? ?
?yf ??yp
g
X ---! B
If B is connected, then:
F ib(E=Y ! B=X) >> F ib(g : X ! B ) ? F ib(p : E ! B )
Proof. By changing the homotopy pull-back diagram of the theorem in a homotopy
meaningful way, we can arrange so that g : X ! B , p : E ! B and gOf : Y ! B *
*are
fibrations and the diagram becomes a pull-back square.
Let G = F ib(g : X ! B ), H = F ib(gOf : Y ! B ) and F = F ib(p : E ! B ). It
is obvious that H = G x F and the induced maps H ! G and H ! F are the
p1 p2
projections. By definition: G ? F = hocolim(G - G x F - ! F ). It follows th*
*at
hocolim(G H ! F ) is connected and by theorem 5.1:
F ib(E=Y ! B=X) >> hocolim(F H ! G) = F ? G
__
|_ |
As a corollary we get Serre's theorem (see [9, theorem 6.1]):
Corollary 5.3 (Serre). Let B be a connected space. Let the following be a hom*
*otopy
pull-back square:
Y --l-! E
?? ?
?yf ??yp
g
X ---! B
Let G = F ib(g : X ! B ) and F = F ib(p : E ! B ). If fHi(G) = ? for i n and
fHi(F ) = ? for i m, then Hi(p; f): Hi(E; Y )! Hi(B; X) is an isomorphism for
i n + m + 2 and an epimorphism for i = n + m + 3.
Proof. According to theorem 5.2, F ib(E=Y ! B=X) >> G ? F . SinceifHi(G ? F )j=*
* ?
for i m + n + 2 (see corollary 4.3), it follows that also fHiF ib(E=Y ! B=X) *
*= ?
i j
for i m + n + 2 (see example 3.3). By corollary 3, fHiCof(E=Y ! B=X) = ? for
i n + m + 3. This is equivalent to the map Hi(p; f): Hi(E; Y )! Hi(B; X) being
an isomorphism for i n + m + 2 and an epimorphism for i = n + m + 3. __|_ |
A GENERALIZATION OF THE TRIAD THEOREM OF BLAKERS-MASSEY 11
6. The case of a cofibration
Let A ! X ! X=A be a cofibration sequence for which A and X are connected.
We will try to find out how far this cofibration sequence is from being a fibra*
*tion
sequence. Let F ! X be the homotopy fiber of the cofiber map X ! X=A. By
proposition 3.6, we already know that F >> A, in particular F is connected. Let
q
q : A ! F be a map such that the composition (A ! F ! X) equals to A ! X.
The map q : A ! F measures the difference between the original cofibration sequ*
*ence
and the fibration sequence F ! X ! X=A. The purpose of this section is to give
some cellular estimation for the homotopy fiber and the homotopy cofiber of the*
* map
q : A ! F .
The above data can be put together by assuming that we have the following com-
mutative diagram:
A P---! P P
? P P P ? P P P id
?? P PqP?? P PP
y y Pq F ---!Pq P
X P---! X=A ?? ??
P P P P P P?yid ?y
PPiPd PP P
Pq X ---Pq!X=A
where:
o P is contractible,
o the following are respectively a homotopy push-out square and a homotopy
pull-back square:
A ---! P F ---! P
?? ? ? ?
?y ??y ??y ??y
X ---! X=A X ---! X=A
Proposition 6.1. q : A ! F has a right homotopy inverse.
Proof. Commutativity of the following diagram proves clearly the proposition:
i j
ff|_ A? ' hocolim ? - A -! ?
| ?? ?? ?? ??
|| yq i ?y ?yq ?y j
id|| F ' hocolim ? - F -! P
| ?? ?? ?? ??
| ? ? ? ?
| yr y y y
|__- i j
A ' hocolim ? - X -! X=A
__
|_ |
12 WOJCIECH CHACHOLSKI
Notation. By r : F ! A we denote the homotopy inverse of q : A ! F
constructed in proposition 6.1.
Remark. Notice that since q : A ! F has a right homotopy inverse, q : A !*
* F
induces a monomorphism on homology. This observation is due to William Dwyer.
Corollary 6.2. (E. Dror Farjoun, see [6]). Closed classes C(A) and C(F ) are
equal.
Proof. According to proposition 3.6, F >> A. By suspending this inequality we g*
*et:
F >> A (see corollary 3.6). This proves C(F ) C(A).
Since the map q : A ! F has a right homotopy inverse, proposition 3.7 im*
*plies
A >> F . This shows C(A) C(F ). __|_ |
We start investigating the map q : A ! F by looking at the homotopy fiber
of its right homotopy inverse r : F ! A .
Proposition 6.3. F ib(r : F ! A ) >> F ? (X=A).
Proof. Let us consider the following homotopy pull-back square:
F - --! P
?? ?
?y ??y
X - --! X=A
The homotopy fiber of P ! X=A is weakly equivalent to (X=A). By applying
theorem 5.2, we get: F ib(r : F ! A ) >> F ? (X=A). __|_ |
The following proposition gives a cellular estimation for F ib(q : A ! F *
* ). Al-
though this inequality is in terms of F , out of it we will be able to extract *
*more
useful and calculable inequalities. Those inequalities are going to be in term*
*s of A
and either the homotopy fiber F ib(A ! X) or the homotopy cofiber X=A.
Proposition 6.4. F ib(q : A ! F ) >> F ^ (X=A).
q r
Proof. The composition (A -! F ! A) is a weak equivalence, thus:
F ib(q : A ! F ) ' F ib(r : F ! A )
Since F is connected (it is A-cellular), so F ? (X=A) is simply connected and a*
*c-
cording to proposition 3.7, we can loop the inequality of proposition 6.3. As a*
* result
we get:
i j
F ib(q : A ! F ) ' F ib(r : F ! A ) >> F ? (X=A)
i j i j
Since F ? (X=A) ' F ^ (X=A) , it follows from proposition 3.7 that:
i j
F ib(q : A ! F ) >> F ^ (X=A) >> F ^ (X=A)
A GENERALIZATION OF THE TRIAD THEOREM OF BLAKERS-MASSEY 13
__
|_ |
The following estimation of the homotopy fiber F ib(q : A ! F ) is entir*
*ely in
terms of A and the homotopy cofiber X=A:
Theorem 6.5. F ib(q : A ! F ) >> A ^ (X=A).
Proof. Since F >> A (see proposition 3.6), using corollary 3.10, we get:
F ib(q : A ! F ) >> F ^ (X=A) >> A ^ (X=A)
__
|_ |
Corollary 6.6. If F ib(A ! X) is connected, then:
F ib(q : A ! F ) >> A ^ F ib(A ! X)
Proof. Since F ib(A ! X) is connected and X=A >> F ib(A ! X) (see proposi-
tion 3.7), according again to proposition 3.7, (X=A) >> F ib(A ! X). Therefore
we can write:
F ib(q : A ! F ) >> A ^ (X=A) >> A ^ F ib(A ! X)
__
|_ |
We continue investigating q : A ! F by giving a cellular estimation for*
* its
homotopy cofiber:
Theorem 6.7. (F=A) >> F ? (X=A).
Proof. According to proposition 3.7:
(F=A) ' Cof(q : A ! F ) >> F ib(q : A ! F )
Proposition 6.4 gives F ib(q : A ! F ) >> F ^ (X=A). By suspending this
inequality we get:
i j
(F=A) >> F ib(q : A ! F ) >> F ^ (X=A) ' F ? (X=A)
__
|_ |
Remark. The inequality (F=A) >> F ib(q : A ! F ), that appears in the
proof of theorem 6.7, can not be usually de-suspended.
By the same argument as in the theorem 6.5 we can get a cellular estimation o*
*f the
homotopy cofiber of q : A ! F in terms of A and the homotopy cofiber X=A:
Corollary 6.8. (F=A) >> A ? (X=A)
Under an extra assumption we can get even more explicit formula:
Corollary 6.9. If X=A is weakly equivalent to the suspension of a connected sp*
*ace,
then (F=A) >> A ^ X=A.
14 WOJCIECH CHACHOLSKI
Proof. Let X=A ' Y . According to proposition 3.7: (X=A) ' Y >> Y . By
suspending this inequality, we get: (X=A) >> Y ' X=A. Since:
i j
(F=A) >> A ? (X=A) ' A ^ (X=A) ' A ^ (X=A)
corollary 3.10 implies: (F=A) >> A ^ X=A. __|_ |
Using the same arguments as in corollary 6.6 one can easily prove the followi*
*ng
cellular estimation of the homotopy cofiber of q. This inequality is in terms *
*of A
and the homotopy fiber of A ! X:
Corollary 6.10. If F ib(A ! X) is connected, then:
(F=A) >> A ? F ib(A ! X)
Out of proven inequalities we can recover Serre's result (see [9, corollary 6*
*.3]):
Corollary 6.11. Let A be simply connected. If:
o fHi(A)i= ? for ij n, i j
o fHj F ib(A ! X) = ? or fHj (X=A) = ? for j m,
then ssi(q): ssi(A)! ssi(Fi)s an isomorphism for i n + m and an epimorphism for
i = n + m + 1.
Proof. Since (F=A) >> A?(X=A) and (F=A) >>iA?F ib(Aj! X), according to
corollary 4.3, the assumptions imply: fHi (F=A) = ? for i n + m + 2. It follo*
*ws
that fHi(F=A) = ? for i n + m + 1.
Since F is A-cellular and A is simply connected, F is also simply connected. *
* It
implies: ss1(F; A) = ?. Using relative Hurewicz theorem (see [8, proposition *
*1.7]),
we get: ssi(F; A) = ? for i n + m + 1. It proves that ssi(q): ssi(A)! ssi(Fi)*
*s an
isomorphism for i n + m and an epimorphism for i = n + m + 1. __|_ |
7. The case of a push-out
In this section we are going to investigate how far a homotopy push-out square
is from being a homotopy pull-back square. Let B be a connected space. Let us
consider the following commutative diagram:
F
?
A P- --! E P ??
? P P Pq? P P Pid y
?? P?P?Pq PP Pq
y y Y - --! E
? ?
G ---! X P- --! B P ?? ??
P P Pid P PyPid y
PP Pq PP Pq
X - --! B
where:
A GENERALIZATION OF THE TRIAD THEOREM OF BLAKERS-MASSEY 15
o the following are respectively a homotopy push-out square and a homotopy
pull-back square:
A ---! E Y ---! E
?? ? ? ?
?y ??y ??y ??y
X ---! B X ---! B
o F ! E is the homotopy fiber of E ! B,
o G ! X is the homotopy fiber of X ! B,
The map q : A ! Y measures how far the above homotopy push-out square is from
being homotopy pull-back. The purpose of this section is to give some cellular *
*esti-
mation for the homotopy fiber and the homotopy cofiber of the map q : A ! Y *
* .
Theorem 7.1. If F or G is connected, then F ib(q : A ! Y ) >> F ^ G
q
Lemma 7.2. Let A ! Y ! E be two composable maps. The following is a homotopy
push-out square:
E=A ---! A
?? ?
?y ??yq
E=Y ---! Y
where
o E=A ! A and E=Y ! Y are the homotopy cofibers respectively of
E ! E=A and E ! E=Y ,
o the map E=A ! E=Y is induced by id : E ! E and q : A ! Y .
Proof. We have to show that there is a weak equivalence:
hocolim(A E=A ! E=Y ) -! Y
whose restriction to A is q : A ! Y . Let us consider the following commuta*
*tive
diagram:
? --- ? - --! ? ?
x? x x x
?? ??? ??? ???
q
A --id- A - --! Y Y
?? ? ? ?
?y ??y ??y ??y
? --- E - id--! E ?
A --- E=A - --! E=Y
16 WOJCIECH CHACHOLSKI
Applying the homotopy colimit functor first to the columns of this diagram and *
*then
to the obtained row, we get: hocolim(A E=Y ! E=Y ). Changing the order
of this procedure and applying the homotopy colimit functor first to the rows o*
*f the
diagram and then to the obtained column, we get: Y . This proves the lemma. __*
*|_ |
According to proposition 3.6, lemma 7.2 implies:
q
Corollary 7.3. Let A ! Y ! E be two composable maps. If F ib(E=A ! E=Y ) is
connected, then: F ib(q : A ! Y ) >> F ib(E=A ! E=Y )
Proof of the theorem.Corollary 7.3 implies that in order to prove the theorem, *
*it is
enough to show that if F or G is connected, then F ib(E=A ! E=Y ) >> F ^ G.
Let us consider the following commutative diagram:
A ---! Y ---! X
? q
?? ?? ??
y ?y ?y
E --id-!E ---! B
This diagram induces maps between the homotopy cofibers E=A ! E=Y ! B=X.
Since the appropriate square is homotopy push-out, the map E=A ! B=X is a weak
equivalence. It implies:
F ib(E=A ! E=Y ) ' F ib(E=Y ! B=X)
Since the appropriate square is homotopy pull-back, theorem 5.2 gives:
F ib(E=Y ! B=X) >> F ? G ' (F ^ G)
If F or G is connected, then F ^ G is also connected and proposition 3.7 implie*
*s:
F ib(E=A ! E=Y ) ' F ib(E=Y ! B=X) >> F ^ G
__
|_ |
If F ib(A ! E) and F ib(A ! X) are connected, then F >> F ib(A ! X) and
G >> F ib(A ! E) (see proposition 3.6). As a consequence we get:
Corollary 7.4. If F ib(A ! E) and F ib(A ! X) are connected, then:
F ib(q : A ! Y ) >> F ib(A ! X) ^ F ib(A ! E)
As it was in the case of a cofibration, we can use cellular estimations for t*
*he ho-
motopy fiber F ib(q : A ! Y ) to get some cellular estimations for the hom*
*otopy
cofiber (Y=A):
Theorem 7.5. If F ib(A ! E) and F ib(A ! X) are connected then:
(Y=A) >> F ib(A ! E) ? F ib(A ! X)
As a corollary we get:
A GENERALIZATION OF THE TRIAD THEOREM OF BLAKERS-MASSEY 17
Corollary 7.6. Let p and s be distinct prime numbers. If F ib(A ! E) has p-tor*
*sion
reduced homology and F ib(A ! X) has s-torsion reduced homology, then q : A ! Y
is a homology isomorphism.
Proof. Since F ib(A ! E) and F ib(A ! X) have respectively p and s-torsion homo*
*l-
ogy, K"unneth theorem (see [8, theorem 10.3]) implies that F ib(A ! E)^F ib(A !*
* X)
is acyclic. Thus F ib(q : A ! B ) is also acyclic, so q : A ! B indu*
*ces an
isomorphism on homology. It follows that q : A ! Y is a homology isomorphism. *
*__|_ |
Out of proven inequalities we can recover the following result of Blakers-Mas*
*sey
(see [9, theorem 7.14]):
Corollary 7.7. Let the following be a homotopy push-out of simply connected sp*
*aces:
A ---! E
?? ?
?y ??y
X ---! B
If Hi(X; A) = ?, for i n (n 1) and Hi(E; A) = ?, for i m (m 1), then
ssi(E; A) ! ssi(B; X) is an isomorphism for i < n + m and an epimorphism for
i n + m.
Proof. Since A, X and E are simply connected, according to Hurewicz theorem (se*
*e [8,
proposition 7.1]), ssi(X; A) = ? for i n and ssi(E; A) = ? for i m. It follow*
*s:
i j
o ssi F ib(A ! X) = ? for i < n,
i j
o ssi F ib(A ! E) = ? for i < m.
Let Y be the homotopy pull-back: holim(X ! B E) and q : A ! Y be the
naturalimap.j Inequality (Y=A) >> F ib(A ! E) ? F ib(A ! X) implies that:
fHi (Y=A) = ? for i n + m. Therefore: fHi(Y=A) = ? for i < n + m.
The assumption: Eiand B are simply-connected,jimpliesithatjthere is an epimor-
phism: ss2(B) ! ss1 F ib(E ! B) , so ss1 F ib(E ! B) is abelian. Thus ss1(Y )*
* is
also abelian and Hurewicz map ss1(Y; A) ! fH1(Y; A) is an isomorphism. As a con*
*se-
quence we get: ssi(Y; A) = ? for i < n + m. This implies that ssi(A) ! ssi(Y ) *
*is an
isomorphism for i < n + m - 1 and an epimorphism for i < n + m.
By looking at the long exact sequences of homotopy groups of pairs (A ! E) and
(Y ! E), we get that ssi(E; A) ! ssi(E; Y ) is an isomorphism for i < n + m and*
* an
epimorphism for i n + m.
Since ssi(E; Y ) ! ssi(B; X) is an isomorphism for all i, ssi(E; A) ! ssi(B; *
*X) is an
isomorphism for i < n + m and an epimorphism for i n + m. __|_ |
18 WOJCIECH CHACHOLSKI
References
1. A. L. Blakers and W. S. Massey, The homotopy group of a triad 1, Ann. of Ma*
*th. 53 1951,
161-205.
2. W. Chacholski, Functors CWA and PA, Ph.D. thesis, Univ. of Notre Dame 1995.
3. W. Chacholski, Closed classes, Proc. to the conf. in Alg. Top. Barcelona, S*
*ummer 1994.
4. W. Chacholski, Homotopy properties of shapes of diagrams, report No. 6, 199*
*3/94, Institut
Mittag-Leffler.
5. E. Dror Farjoun, Cellular spaces, preprint.
6. E. Dror Farjoun, Cellular inequalities, Proc. to the conf. in Alg. Top. Nor*
*theastern Univ. June
1993, Springer Verlag.
7. I. Namioka, Maps of pairs in homotopy theory, Proc. London Math. Soc. 12 19*
*62, 725-738.
8. E. H. Spanier, Algebraic Topology, McGraw-Hill 1964.
9. G. W. Whitehead, Elements of Homotopy Theory, Grad. Texts in Math. 61, Spri*
*nger 1978.
Wojciech Chacholski, Department of Mathematics, University of Notre Dame,
Mail Distribution Center, Notre Dame, Indiana 46556-5683
E-mail address: wchacho1@kenna.math.nd.edu