топология / Tom Dieck T. Algebraic topology (EMS, 2008)
.pdf158 Chapter 6. Homotopy Groups
11.Determine 2n 1.Sn _ Sn/ for n 2.
12.Let fj be a self-map of Sn.j /. Show d.f1 ? f2/ D d.f1/d.f2/.
13. Let H W 3.S2/ ! Z be the isomorphism which sends (the class of the) Hopf mapW S3 ! S2 to 1 (the Hopf invariant). Show that for f W S3 ! S3 and g W S2 ! S2 the relations H.˛ ı f / D d.f /H.˛/ and H.g ı ˛/ D d.g/2H.˛/ hold.
Chapter 7
Stable Homotopy. Duality
The suspension theorem of Freudenthal indicates that homotopy theory simplifies by use of iterated suspensions. We use this idea to construct the simplest stable homotopy category. Its construction does not need extensive technical considerations, yet it has interesting applications. The term “stable” refers to the fact that iteration of suspension induces after a while a bijection of homotopy classes.
We use the stable category to give an introduction to homotopical duality theory. In this theory the stable homotopy type of a closed subspace X Rn and its complement Rn X X are compared. This elementary treatment of duality theory is based on ideas of Albrecht Dold and Dieter Puppe; see in particular [54]. It is related to the classical Alexander duality of homology theory and to Spanier–Whitehead duality.
We introduced a naive form of spectra and us them to define spectral homology and cohomology theories. The homotopical Euclidean complement duality is then used to give a simple proof for the Alexander duality isomorphism. In a later chapter we reconsider duality theory in the context of product structures.
7.1 A Stable Category
Pointed spaces X and Y |
are called stably homotopy equivalent, ink |
k's |
Y , |
symbols X |
|
if there exists an integer k 0 such that the suspensions † X and † Y are
homotopy equivalent. Pointed maps |
f; g W X |
! Y are called |
stably homotopic, |
|||||
|
|
k |
|
|
k |
|
||
in symbols f 's g, if for some integer k the suspensions † |
|
f |
and † |
|
g are |
|||
homotopic. We state some of the results to be proved in this chapter which use these notions.
(7.1.1) Theorem (Stable Complement Theorem). Let X and Y be homeomorphic closed subsets of the Euclidean space Rn. Then the complements Rn XX and Rn XY are either both empty or they have the same stable homotopy type with respect to arbitrary base points.
In general the complements themselves can have quite different homotopy type. A typical example occurs in knot theory, the case that X Š Y Š S1 are subsets of R3. On the other hand the stable homotopy type still carries some interesting geometric information: see (7.1.10).
(7.1.2) Theorem (Component Theorem). Let X and Y be closed homeomorphic subsets of Rn. Then 0.Rn X X/ and 0.Rn X Y / have the same cardinality.
160 Chapter 7. Stable Homotopy. Duality
Later we give another proof of Theorem (7.1.2) based on homology theory, see (10.3.3). From the component theorem one can deduce classical results: The Jordan separation theorem (10.3.4) and the invariance of domain (10.3.7).
Theorem (7.1.1) is a direct consequence of (7.1.3). One can also compare complements in different Euclidean spaces. The next result gives some information about how many suspensions suffice.
(7.1.3) Theorem. Let X Rn and Y Rm be closed subsets and h W X ! Y a homeomorphism. Suppose n m. Then the following holds:
(1)If Rn 6D X, then Rm 6D Y , and h induces a canonical homotopy equivalence
†mC1.Rn X X/ ' †nC1.Rm X Y / with respect to arbitrary base points.
(2)If Rn D X and Rm 6D Y , then n < m and †nC1.Rm X Y / ' Sm, i.e., Rm X Y has the stable homotopy type of Sm n 1.
(3)If Rn D X and Rm D Y , then n D m.
In many cases the number of suspensions is not important. Since it also depends on the situation, it is convenient to pass from homotopy classes to stable homotopy classes. This idea leads to the simplest stable category.
The objects of our new category ST are pairs .X; n/ of pointed spaces X and integers n 2 Z. The consideration of pairs is a technical device which allows for a better formulation of some results. Thus we should comment on it right now.
The pair .X; 0/ will be identified with X. The subcategory of the objects
.X; 0/ D X with morphisms the so-called stable homotopy classes is the geometric input. For positive n the pair .X; n/ replaces the n-fold suspension †nX. But it will be convenient to have the object .X; n/ also for negative n (“desuspension”). Here is an interesting example. In the situation of (7.1.3) the homotopy equivalence †mC1.Rn X X/ ! †nC1.Rm X Y / induced by h represents in the category ST an isomorphism h W .Rn X X; n/ ! .Rm X Y; m/. In this formulation it then makes sense to say that the assignment h 7!h is functor. (Otherwise we would have to use a mess of different suspensions.) Thus if X is a space which admits an embedding i W X ! Rn as a proper closed subset for some n, then the isomorphism type of .Rn X i.X/; n/ in ST is independent of the choice of the embedding. Hence we have associated to X a “dual object” in ST (up to canonical isomorphism).
|
t |
Let †t X D X ^ St be the t-fold suspension oft |
X. As a model for the sphere |
|
S |
|
we uset eithert |
the one-point compactification R |
[ f1g or the quotient space |
S.t/ D I =@I . |
In these cases we have a canonical associative homeomorphism |
|||
Sa ^ Sb Š SaCb which we usually treat as identity. Suppose n; m; k 2 Z are integers such that n C k 0; m C k 0. Then we have the suspension morphism
† |
W |
ŒX |
^ |
SnCk ; Y |
^ |
SmCk 0 |
! |
ŒX |
^ |
SnCkC1 |
; Y |
^ |
SmCkC1 |
0; f |
f |
id.S1/: |
|
|
|
|
|
|
|
|
|
7! ^ |
|
We form the colimit over these morphisms, colimk ŒX ^ SnCk ; Y ^ SmCk 0. For n C k 2 the set ŒX ^ SnCk ; Y ^ SmCk 0 carries the structure of an abelian group
7.1. A Stable Category |
161 |
and † is a homomorphism. The colimit inherits the structure of an abelian group. We define as morphism group in our category ST
ST..X; n/; .Y; m// D colimk ŒX ^ SnCk ; Y ^ SmCk 0:
Formation of the colimit means the following: An element of ST..X; n/; .Y; m// is represented by pointed maps fk W X ^ SnCk ! Y ^ SmCk , and fk , fl ; l k represent the same element of the colimit if †l k fk ' fl . Composition of morphisms is defined by composition of representatives. Let fk W X ^ SnCk ! Y ^ SmCk and gl W Y ^ SmCl ! Z ^ SpCl be representatives of morphisms and let r k; l. Then the following composition of maps represents the composition of the morphisms (dotted arrow):
X ^ S |
C |
|
D X ^ S |
C |
|
^ S |
|
|
|
†r k fk |
Y ^ S |
|
C |
|
^ S |
|
|
n |
|
r |
n |
|
k |
r |
|
k |
|
|
|
m |
|
k |
r |
|
k |
|
|
|
|
|
|
|
|
|
|
†r l gl |
|
|
|
D |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
Z ^ SpCr D Z ^ SpCl ^ Sr l |
|
Y ^ SmCl ^ Sr l : |
|||||||||||||||
One verifies that this definition does not depend on the choice of representatives. The group structure is compatible with the composition
ˇ ı .˛1 C ˛2/ D ˇ ı ˛1 C ˇ ı ˛2; .ˇ1 C ˇ2/ ı ˛ D ˇ1 ı ˛ C ˇ2 ı ˛:
The category ST has formal suspension automorphisms †p W ST ! ST, p 2 Z
.X; n/ 7!.X; n C p/; f 7!†pf:
If f W .X; n/ ! .Y; m/ is represented by fk W †nCk X ! †mCk Y m C k 0, k jpj), then †pf is represented by
.†pf /k D †p.fk / W †nCkCpX ! †mCkCpY;
.†pf /kCjpj D fk W †nCkCpCjpjX ! †mCkCpCjpjY;
(with nC k 0,
p 0; p 0:
The rules †0 D id.ST/ and †p ı †q D †pCq show that †p is an automorphism. For p > 0 we call †p the p-fold suspension and for p < 0 the p-fold desuspension.
We have a canonical isomorphism p W .X; n/ ! .†pX; n p/; it is represented by the identity X ^ SnCk ! .X ^ Sp/ ^ SnCk p for n C k p 0. We write
X for the object .X; 0/. Thus for positive n the object .X; n/ can be replaced by
†nX.
(7.1.4) Example. Pointed spaces X; Y are stably homotopy equivalent if and only if they are isomorphic in ST. The image ST.f / of f W X ! Y in ST.X; Y / is called homotopic if and
the stable homotopy class of f . Maps f; g W X ! Y are stably |
k |
0 |
|
||
only if they represent the same element in ST.X; Y /. The groups ST.S |
; S |
|
/ D |
||
colimn nCk .S |
n |
/ are the stable homotopy groups of the spheres. |
|
||
|
|
|
Þ |
||
162 Chapter 7. Stable Homotopy. Duality
(7.1.5) Example. It is in general difficult to determine morphism groups in ST. But we know that the category in non-trivial. The suspension theorem and the degree
theorem yield
ST.Sn; Sn/ D colimk ŒSnCk ; SnCk 0 Š Z:
The composition of morphisms corresponds to multiplication of integers. |
Þ |
(7.1.6) Proposition. Let Y be pathwise connected. We have the embedding i W Sn ! Y C ^ Sn, x 7!. ; x/ and the projection p W Y C ^ Sn ! Sn, .y; x/ 7!x with pi D id. They induce isomorphisms of the k -groups for k n 1.
Proof. Let n D 1. Then Y C ^ S1 Š Y S1=Y f g is path connected. The base point of Y C is non-degenerate. Hence the quotient †0.Y C/ ! †.Y C/ from the unreduced suspension to the reduced suspension is an h-equivalence. The projection Y ! P onto a point induces a 2-connected map between double mapping cylinders
†0.Y C/ D Z. Y C f g ! / ! Z0. P C f g ! / D †0.P C/ Š S1:
From this fact one deduces the assertion for n D 1.
We now consider suspensions |
|
|
|
|
||||||||
k .Sn/ |
|
i |
|
k .Y C ^ Sn/ |
|
p |
k .Sn/ |
|||||
|
|
|
|
|
|
|||||||
|
† |
|
|
|
†Y |
|
|
|
† |
|||
|
|
i |
|
|
|
p |
|
|
|
|||
kC1.SnC1/ |
kC1.Y C ^ SnC1/ |
kC1.SnC1/: |
||||||||||
|
|
|||||||||||
The vertical morphisms are bijective (surjective) for k 2n 2 (k D 2n 1). For n D 1 1.Y C ^ S1/ Š Z. Since 2.Y C ^ S2/ contains 2.S2/ Š Z as a direct summand, we conclude that †Y is an isomorphism. For n 2 we can use directly
the suspension theorem (6.10.4). |
|
||||||
(7.1.7) Proposition. |
Let .Yj |
j j 2 J / be the family of path components of Y |
and |
||||
|
j |
W Yj ! Y |
|
||||
c |
|
the inclusion. Let n 2. Then |
|
||||
|
|
|
h cj i W |
L |
|
||
|
|
|
|
k .YjC ^ Sn/ ! k .Y C ^ Sn/ |
|
||
|
|
|
|
|
j 2J |
|
|
is an isomorphism for 0 k n. In particular n.Y C ^ Sn/ is a free abelian group of rank j 0.Y /j.
Proof. (7.1.6) and (6.10.6). |
|
|
|
|
|
(7.1.8) Proposition. Let Y be well-pointed and n |
|
2. Then |
Sn/ is |
||
|
|
n.Y ^ n |
/ |
! |
|
a free abelian group of rank j 0.Y /j 1 and the suspension n.Y ^ S |
|||||
nC1.Y ^ SnC1/ is an isomorphism.
7.1. A Stable Category |
163 |
From the exact homotopy sequence of the pair .Y C ^ Sn; Sn/ we conclude thatk .Y C ^ Sn; Sn/ D 0 for 0 < k < n. The quotient theorem (6.10.2) shows thatk .Y C ^ Sn; Sn/ ! k .Y C ^ Sn=Sn/ Š k .Y ^ Sn/ is bijective (surjective) for 0 < k 2n 2 (k D 2n 1). From the exact sequence 0 ! n.Sn/ !n.Y C ^Sn/ ! n.Y C ^Sn; Sn/ ! 0 we deduce a similar exact sequence where the relative group is replaced by n.Y ^ Sn/. The inclusion of n.Sn/ splits. Now we can use (7.1.7).
(7.1.9) Corollary. Let X be a well-pointed space. Then ST.S0; X/ is a free abelian group of rank j 0.X/j 1. Þ
The group ST.S0; X/ only depends on the stable homotopy type of X. Therefore we can state:
(7.1.10) Corollary. Let X and Y be well-pointed spaces of the same stable homotopy type. Then j 0.X/j D j 0.Y /j. Therefore (7.1.2) is a consequence of (7.1.3).
Þ
The category ST has a “product structure” induced by the smash product. The category ST together with this additional structure is called in category theory a symmetric tensor category (also called a symmetric monoidal category). The tensor product of objects is defined by
.X; m/ ˝ .Y; n/ D .X ^ Y; m C n/:
Let
fk W X ^ SmCk ! X0 ^ Sm0 Ck ; gl W Y ^ SnCl ! Y 0 ^ Sn0 Cl |
! |
|||||||
be representing maps for morphisms f |
W |
.X; m/ |
! |
.X0; m0/ |
and g |
W |
0 |
|
|
|
|
.Y; n/ |
|
||||
.Y 0; n0/. A representing morphism .f ˝ g/kCl is defined to be . 1/k.nCn / times |
|||||||||||||||||||||
the composition 0 ı .fk ^ gl / ı (dotted arrow) |
|
|
|
|
|
|
|
|
|
||||||||||||
X ^ Y ^ S |
|
C |
C |
C |
|
|
|
X ^ S |
|
C |
|
^ Y ^ S |
|
C |
|
||||||
|
m |
|
k |
n |
|
|
l |
|
|
|
|
|
m |
|
k |
|
fk ^gl |
n |
|
|
l |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
0 |
|
|
0 |
|
|
0 |
|
|
|
0 |
|
|
|
|
|
0 |
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||
X0 ^ Y 0 ^ Sm |
CkCn |
Cl |
|
X0 |
^ Sm |
Ck ^ Y 0 ^ Sn |
Cl |
||||||||||||||
where and 0 interchange two factors in the middle. Now one has to verify:
(1) The definition does not depend on the representatives; (2) the functor property
.f 0 ˝ g0/.f ˝ g/ D f 0f ˝ g0g holds; (3) the tensor product is associative. These
requirements make it necessary to introduce signs in the definition. The neutral object is .S0; 0/. The symmetry c W .X; m/ ˝ .Y; n/ ! .Y; n/ ˝ .X; m/ is . 1/mn
times the morphism represented by the interchange map X ^ Y ! Y ^ X.
164 Chapter 7. Stable Homotopy. Duality
Problems
1.The spaces S1 S1 and S1 _ S1 _ S2 are not homotopy equivalent. They have different fundamental group. Their suspensions are homotopy equivalent.
2.The inclusion X Y ! XC Y induces for each pointed space Y a homeomorphism
.X Y /=.X f g/.
3.Let X and Y be well-pointed spaces. Then Y ! .X Y /=.X f g/, y 7!. ; y/ is a cofibration.
4.Let P be a point. We have an embedding P C ^ Y ! XC ^ Y and a canonical homeomorphism X ^ Y ! XC ^ Y =P C ^ Y .
7.2 Mapping Cones
We need a few technical results about mapping cones. Let f W X ! Y be a pointed map. We use as a model for the (unpointed) mapping cone C.f / the double mapping
cylinder Z.Y |
X ! |
/; it is the quotient of Y C X I C f g under the relations |
f .x/ .x; 0/; .x; 1/ |
. The image of is the basepoint. For an inclusion |
|
W A X we write C.X; A/ D C. /. For empty A we have C.X; ;/ D XC. Since we will meet situations where products of quotient maps occur, we work in the category of compactly generated spaces where such products are again quotient maps. The mapping cone is a functor C W TOP.2/ ! TOP0; a map of pairs
.F; f / W .X; A/ ! .Y; B/ induces a pointed map C.F; f / W C.X; A/ ! C.Y; B/, and a homotopy .Ft ; ft / induces a pointed homotopy C.Ft ; ft /. We note for further use a consequence of (4.2.1):
(7.2.1) Proposition. If F and f |
are h-equivalences, then C.F; f / is a pointed |
|||||||||||||||||||||||||
h-equivalence. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
(7.2.2) Example. |
We write C n |
D |
C. |
R |
n; |
R |
n |
X |
0/. This space will be our model for |
|||||||||||||||||
|
n |
|
|
|
|
|
|
|
|
|
|
|
n |
! S |
n |
, we ob- |
||||||||||
the homotopy type of S |
|
. In order to get a homotopy equivalence C |
|
|
||||||||||||||||||||||
serve that Sn is homeomorphic to the double mapping cylinder Z. |
|
Sn 1 |
! |
/. |
||||||||||||||||||||||
We have the canonical projection from C |
n |
|
|
|
Z.R |
n |
R |
n |
|
0 |
|
|
|
|
|
|
||||||||||
|
|
x |
|
|
|
|
X |
! |
/. An explicit |
|||||||||||||||||
|
|
|
|
|
|
|
|
|
|
D |
|
|
|
|
|
|
|
|
|
|
Þ |
|||||
homotopy equivalence is .x; t/ 7!.sin t |
|
; cos t/, x 7!.0; : : : ; 0; 1/. |
|
|
||||||||||||||||||||||
kxk |
|
|
||||||||||||||||||||||||
(7.2.3) Example. Let X Rn be a closed subspace. Then |
|
|
|
|
|
|
|
|
|
|||||||||||||||||
C.Rn; Rn XX/ D Z.Rn |
Rn XX ! / ' Z. |
|
Rn XX ! / D †0.Rn XX/; |
|||||||||||||||||||||||
the unpointed suspension. If X D Rn, then this space is h-equivalent to S0. If
6D |
X |
n |
X |
X/ is |
X Rn, then Rn |
|
X is well-pointed with respect to any point and †0.Rn |
|
|
h-equivalent to the pointed suspension †.R X X/. |
|
Þ |
||
We are mainly interested in the homotopy type of C.X; A/ (under f |
g C X). |
It is sometimes convenient to provide the set C.X; A/ with a possibly |
different |
7.2. Mapping Cones |
165 |
topology which does not change the homotopy type. Set theoretically we can view
C.X; A/ as the quotients C1.X; A/ D .X 0 [ A I /=A 1 or C2.X; A/ D
.X 0 [ A I [ X 1/=X 1. We can provide C1 and C2 with the quotient topology. Then we have canonical continuous maps p W C.X; A/ ! C1.X; A/ and q W C1.X; A/ ! C2.X; A/ which are the identity on representative elements.
(7.2.4) Lemma. The maps p and q are homotopy equivalences under f g C X. Proof. Define pN W C1.X; A/ ! C.X; A/ by
pN.x; t/ D x; t 1=2; pN.a; t/ D .a; max.2t 1; 0//; pN.a; 1/ D :
One verifies that this assignment is well-defined and continuous. A homotopy ppN ' id is given by ..x; t/; s/ 7!.x; st C .1 s/ max.2t 1; 0//. A similar formula works for ppN ' id. Define qN W C2.X; A/ ! C1.X; A/ by
qN.x; t/ D .x; min.2t; 1//; t < 1; qN.x; t/ D D fA 1g; t 1=2:
Again linear homotopies in the t-coordinate yield homotopies from qqN and qqN to the identity.
(7.2.5) Proposition (Excision). Let U A X and suppose there exists a functionW X ! I such that U 1.0/ and 1Œ0; 1Œ A. Then the inclusion of pairs induces a pointed h-equivalence g W C.X X U; A X U / ! C.X; A/.
Proof. Set .x/ D max.2 .x/ 1; 0/. A homotopy inverse of g is the map f W .x; t/ 7!.x; .x/t/. The definition of f uses the notation C2 for the mapping cone. The homotopies from fg and gf to the identity are obtained by a linear homotopy in the t-coordinate.
(7.2.6) Remark. Mapping cones of inclusions are used at various occasions to relate the category TOP.2/ of pairs with the category TOP0 of pointed spaces. We make some general remarks which concern the relations. They will be relevant for the investigation of homology and cohomology theories.
h |
TOP0 |
|
! C be a homotopy invariant functor. We define an associated |
|||
Let Q W |
|
h |
|
|||
functor h |
|
P Q |
|
TOP.2/ |
|
C by composition with the mapping cone functor |
7! |
|
|
|
|
|
|
.X; A/ |
C.X; A/. The functor P Q |
|||||
|
D |
|
W |
|
! |
h is homotopy invariant in a stronger sense: If |
f W .X; A/ ! .Y; B/ is a map of pairs such that the components f W X ! Y and f W A ! B are h-equivalences, then the induced map h.X; A/ ! h.Y; B/ is an isomorphism (see (7.2.1)). Moreover h satisfies excision: Under the hypothesis of (7.2.5) the inclusion induces an isomorphism h.X X U; A X U / Š h.X; A/.
Conversely, let h W TOP.2/ ! C be a functor. We define an associated functor
Rh D Q |
D |
h.X; / and with the obvious induced morphisms. |
h on objects by Rh.X/ |
|
If h is homotopy invariant, then also Rh.
166 Chapter 7. Stable Homotopy. Duality
The composition PR is given by PRh.X; A/ D h.C.X; A/; /. We have natural morphisms
h.C.X; A/; / ! h.C.X; A/; CA/ h.C.X; A/ X U; CA X U / h.X; A/:
Here CA is the cone on A and U CA is the subspace with t-coordinates in Œ1=2; 1 . If h is strongly homotopy invariant and satisfies excision, then these morphisms are
isomorphisms, i.e., PR is naturally isomorphic to the identity. |
|
h.X/ |
h.C.X; //. There is a canonical |
The composition RP is given by RP Q |
D Q |
projection C.X; / ! X. It is a pointed h-equivalence, if the inclusion f g ! X |
|
h is homotopy invariant, the composition RP is naturally |
|
is a cofibration. Thus if Q |
Þ |
isomorphic to the identity on the subcategory of well-pointed spaces. |
|
Let .X; A/ and .Y; B/ be two pairs. We call A Y; X B excisive in X Y if the canonical map p W Z.A Y A B ! X B/ ! A Y [ X B is a homotopy equivalence.
(7.2.7) Proposition (Products). Let .A Y; X B/ be excisive. Then there exists a natural pointed homotopy equivalence
˛ W C.X; A/ ^ C.Y; B/ ! C..X; A/ .Y; B//:
It is defined by the assignments
.x; y/ 7!.x; y/;
.a; s; y/ 7!.a; y; s/;
.x; b; s/ 7!.x; b; s/;
.a; s; b; t/ 7!.a; b; max.s; t//:
(See the proof for an explanation of notation).
Proof. In the category of compactly generated spaces C.X; A/ ^ C.Y; B/ is a quotient of
X Y C A I Y C X B I C A I B I
under the following relations: .a; 0; y/ .a; y/, .x; b; 0/ .x; b/, .a; 0; b; t/
.a; b; t/, .a; s; b; 0/ .a; s; b/, and A 1 B I [ A I B 1 is identified to a base point .
In a first step we show that the smash product is homeomorphic to the double
p
mapping cylinder Z.X Y Z ! f g/ where
Z D Z.A Y A B ! X B/:
This space is the quotient of
X Y C .A Y C A B I C X B/ I C A B .I I=I 0/
7.2. Mapping Cones |
167 |
under the following relations: .x; b; 0/ .x; b/, .a; y; 0/ .a; y/, .a; b; t; 0/
.a; b/, .a; b; 1; s/ .a; b; s/, and .A Y C A B I C X B/ 1 is identified to a base point .
The assignment |
|
|
|
|
|
|
I I ! I I; .u; v/ 7!( |
.2uv; v/; |
u//; |
u |
|
1=2; |
|
.v; 2v.1 |
|
u |
1=2: |
|||
|
|
|
|
|
|
|
induces a homeomorphism 0 W I I=.I 0/ ! I I . Its inverse ˇ0 has the form
I |
|
I |
.0; 0/ |
g ! |
I |
|
.I |
0 |
g |
/; .s; t/ |
7!( |
.1 t=2s; s/; |
s |
t; |
|
|
|
X f |
|
|
X f |
|
.s=2t; t/; |
s |
|
t: |
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
A homeomorphism ˇ W C.X; A/ ^ C.Y; B/ ! C. p/ is now defined by ˇ.x; y/ D
.x; y/, ˇ.a; s; y/ D .a; y; s/, ˇ.x; b; t/ D .x; b; t/, ˇ.a; s; b; t/ D .a; b; ˇ0.s; t//. The diagram
X Y |
|
p |
f g |
||||||
|
Z |
||||||||
|
|
|
|
|
|
|
|
||
|
D |
|
|
p |
|
|
D |
||
|
|
|
|
|
|||||
|
|
|
|
|
|
|
|
||
|
|
|
|||||||
X Y |
|
A Y [ X B |
|
f g |
|||||
induces W C. p/ ! C. /. It is a pointed h-equivalence if p is an h-equivalence. One verifies that ˛ D ˇ.
(7.2.8) Remark. The maps ˛ are associative: For three pairs .X; A/; .Y; B/; .Z; C / the relation ˛.˛ ^ id/ D ˛.id ^˛/ holds. They are also compatible with the interchange map. Finally, they yield a natural transformation. Þ
Problems
1.Verify that the map f in the proof of 7.2.5 is continuous. Similar problem for the homotopies.
2.Let .F; f / W .X; A/ ! .Y; B/ be a map of pairs. If F is n-connected and f .n 1/- connected, then C.F; f / is n-connected.
3.Let X D A [ B and suppose that the interiors Aı; Bı still cover X. Then the inclusion induces a weak homotopy equivalence C.B; A \ B/ ! C.X; A/.
4.Construct explicit h-equivalences C n ! Rn [ f1g D S.n/ such that
C m ^ C n |
|
˛ |
C mCn |
|||||
|
|
|||||||
|
|
Š |
|
|
|
|
||
|
|
|
||||||
S.m/ ^ S.n/ |
S.mCn/ |
|||||||
|
||||||||
is homotopy commutative.