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19 Spectral Unreduced Homology and Cohomology Theories
Spectral Unreduced Homology and Cohomology Theories
By a simple transformation we can transfer the domain of our theories from CS* to (38'. Homology and cohomology theories defined on pairs ( X , A ) are called unreduced homology and cohomology theories (sometimes the adjective unreduced is dropped). We define unreduced theories here and develop their properties on the category E of pairs in CG with the AHEP. In the next section we will consider unreduced theories on more general pairs.
Definition 19.1 For (A',A ) E CQ2 we set E"(X, A ) = Eyx u CA),
Em(X,A ) = &(X u CA),
where the vertex of the cone is chosen as base point. If A = @ we interpret X u C A to mean X with a disjoint point added and used as base point. Em and Em are called the unreduced homology and cohomology theories associated with the spectrum {En}.
Definition 19.2 Let C be a category of pairs of topological spaces. An unreduced homology (cohomology) theory defined on C is a sequence of covariant (contravariant) functors Em (Em) for m E Z satisfying the axioms: (A) (Hornotopy) Letf, g : ( X , A ) -+ ( Y , B). Supposef- g in C (i.e., there is a map H : ( X x I , A x I ) -P ( Y , B ) in C with H(x, 0) = f ( x ) , H(x, 1) = g(x)). Then
f,,= g* : Em(X,A ) + Em(Y , B )
and 183
f* = g* : Em(Y, B) + E"(X, A ) .
184
19. Spectral Unreduced Homology and Cohomology Theories
(B) (Excision) If U is open and D c Int A , the inclusion e : ( X - U, A - U)-+ ( X , A ) induces isomorphisms in homology and cohomology. Abbreviate Em(X,0) and Em(X, 0) as Em(X)and E"(X). (C) (Exactness) There are natural transformations 8: Em(X, A ) + Em-,(A) and 6 : Em(A)-+ Emt'(X,A ) which fit into exact sequences **. 3
Em(A)-+ Em(X)-+ Em(X,A)+
d
Em-,(A) + ' . *
... t E " ( A ) t E " ( X ) t E " ( X , A ) 4 - E m - 1 ( A ) t * * * . d
For the homology and cohomology theories we construct, we will prove stronger excision properties than axiom (B). There are two types of strengthenings of axiom (B) : Type I excision If to be open).
D c Int A , e induces isomorphisms ( U is not assumed
Type 2 excision If U is open and
D c A , e induces isomorphisms.
As we will see in section 21, type 1 excision is natural for homology and type 2 excision is natural for cohomology.
Lemma 19.3 Type 1 excision is equivalent to the condition that if ( X , , X , ) is excisive in X , Em(X, u X , XI) 9
Em(X2
9
XI n X2>
and Em(Xl u X , , X I ) E"(X,, X , n X , ) . Proof To prove B, let X , = A and X , = X - U. Then Int X , u Int X , = Int A u X - B = X . Conversely, if B holds, making the same substitutions we see that D = X - Int X , c Int X , = Tnt A .
Definition 19.4 Let E be the atcegory of pairs ( X , A ) E CS2 with the AHEP. Let N = E n CS* be the category of well-pointed spaces. Theorem 19.5 On G, Emand Emare unreduced homology and cohomology theories.
-
Proof Iff: ( X , A ) ( Y , B ) define X u CA -+ Y u CB by f ( x ) = f ( x ) and f(a, t ) = ( f ( a ) ,t ) . y i s base point preserving. Furthermore, iff g in G, f- in CS*. Since X u CA N X / A , Em(X, u X , , X , ) g Em(X,, X, n),'A and Em(Xl u X , , X , ) r E m ( X 2 , X, n X,) with no hypothesis, for X , u X 2 / X 1 -+
19. Spectra[ Unreduced Homology and Cohomology Theories
185
= X,/X, n X,. (By Exercise 19, Section 14, XI is closed.) Exactness follows immediately from the homeomorphism
x+ u C*A+ = X U CA and 18.9.
I
Corollary 19.6
Em(X)Z Em(X,*) 0 Em(*),
E m ( X )g Em(X,*) @ Em(*).
Proof The exact sequences -+
-.
*
Em(*)+ E,,,(X) -+ Em(X,*) + . * * t E"(X, *)t
+ Ern(*)+ E"(X)
split since there is a map px: X 4 * with px i = 1 * by Exercise 11, Section 11 and 11.11. I Proposition 19.7 If X E A', Em(X, *) g E m ( X ) and Em(X, *) z E m ( X ) . Hence Em(X)E Em(X)0 Em(*)
and
E m ( X )E E m ( X )@ E m ( * ) .
Proof Let X * = X u C*. Then Em(X, *) =Em(X*) and E m ( X , *) = E m ( X * ) .It is sufficient to show that if X E N', ( X , *) N (X*, 1). There is an
obvious map a : X * + X in eG*, and since X E N there is a retraction P : X x / - + X * . L e t y ( x ) = P ( x , l ) . T h e n y : X + K * i s i n C G * . Nowap: 1 ay. Consider H : X* x I+ X x Z defined by H(x, s) = ( x , s), H(t, s) = (*, s + t(1 - s)). Then P H : 1 ya and hence ( X * , 1) N ( X , *). The second statement follows by applying 19.6. I
-
Corollary 19.8 Em(*)g Em(So)z E-"(*).
I
Corollary 19.9 If X E N,
Proposition 19.10 I f X is a CW complex, A m ( X ;n) LZ fi,,(X; n) = 0 for m<0. Proof X A K ( q n) is a CW complex with all cells in dimensions n and larger, except for 0-cells. Hence n m + , , ( X K(T, ~ n ) ) = 0 for m < 0 and n > 1.
19. Spectral Unreduced Homology and Cohomology Theories
186
Thus R J X ; n ) = 0 for m < 0. Since Sm-"Xhas all cells in dimensions n - m and larger, except for 0-cells, [Sn-mX,K(n, n)] = 0 if m < 0 by 16.3 applied to the diagram
The Eilenberg-Steenrod axioms on a positively graded functor are the laws: (A), (B), and (Cj of 19.2 together with the dimension axiom:
(D) A functor satisfying these properties is called an ordinary homology or cohomology theory. They are characteristic properties and are often used as a starting point for making calculations with ordinary homology and cohomology. By 19.5, 19.9, and 19.10 ordinary spectral homology and cohomology satisfy the Eilenberg-Steenrod axioms on the category of CW pairs.
Exercises
1." Prove that if X * * *
* * .
4
+-
3
A
3
B, there are long exact sequences
&(A, B ) -+ Em(X,B ) -+ Em(X,A )
A Em-,(A,B )
-+ *
*.
d
Ern(A,B) t E"(X, B ) t E y X , A ) t E m - l ( A , B ) 4- f . . .
(Compare with Exercise 5, Section 10.) 2. Let En ( E n )be an unreduced homology (cohomology) theory. Define En(X) = En(X, *) (i?"(X) = E"(X, *)). Show that ,!?,,(I?") is a reduced homology (cohomology) theory with the modification that S X is replaced by C X in Axiom (B) and X u C*A is replaced by X u CA in Axiom (C).
3. Show that E n ( u X a )2 0 En(X,), and if X , are CW complexes and E is an 0-spectrum, E"(UX,) r n E " ( X , ) . The assumption that the X , are CW complexes may be dropped if we assume that En is a CW complex for each n (see footnote 15). 4. Let { U,} be an open cover of X such that for all LY, LY' there exists LY" with U , u U,. c U a - .Show that E n ( X )E lim En(U,). (26.22, 26.28)
By a simple transformation we can transfer the domain of our theories from CS* to (38'. Homology and cohomology theories defined on pairs ( X , A ) are called unreduced homology and cohomology theories (sometimes the adjective unreduced is dropped). We define unreduced theories here and develop their properties on the category E of pairs in CG with the AHEP. In the next section we will consider unreduced theories on more general pairs.
Definition 19.1 For (A',A ) E CQ2 we set E"(X, A ) = Eyx u CA),
Em(X,A ) = &(X u CA),
where the vertex of the cone is chosen as base point. If A = @ we interpret X u C A to mean X with a disjoint point added and used as base point. Em and Em are called the unreduced homology and cohomology theories associated with the spectrum {En}.
Definition 19.2 Let C be a category of pairs of topological spaces. An unreduced homology (cohomology) theory defined on C is a sequence of covariant (contravariant) functors Em (Em) for m E Z satisfying the axioms: (A) (Hornotopy) Letf, g : ( X , A ) -+ ( Y , B). Supposef- g in C (i.e., there is a map H : ( X x I , A x I ) -P ( Y , B ) in C with H(x, 0) = f ( x ) , H(x, 1) = g(x)). Then
f,,= g* : Em(X,A ) + Em(Y , B )
and 183
f* = g* : Em(Y, B) + E"(X, A ) .
184
19. Spectral Unreduced Homology and Cohomology Theories
(B) (Excision) If U is open and D c Int A , the inclusion e : ( X - U, A - U)-+ ( X , A ) induces isomorphisms in homology and cohomology. Abbreviate Em(X,0) and Em(X, 0) as Em(X)and E"(X). (C) (Exactness) There are natural transformations 8: Em(X, A ) + Em-,(A) and 6 : Em(A)-+ Emt'(X,A ) which fit into exact sequences **. 3
Em(A)-+ Em(X)-+ Em(X,A)+
d
Em-,(A) + ' . *
... t E " ( A ) t E " ( X ) t E " ( X , A ) 4 - E m - 1 ( A ) t * * * . d
For the homology and cohomology theories we construct, we will prove stronger excision properties than axiom (B). There are two types of strengthenings of axiom (B) : Type I excision If to be open).
D c Int A , e induces isomorphisms ( U is not assumed
Type 2 excision If U is open and
D c A , e induces isomorphisms.
As we will see in section 21, type 1 excision is natural for homology and type 2 excision is natural for cohomology.
Lemma 19.3 Type 1 excision is equivalent to the condition that if ( X , , X , ) is excisive in X , Em(X, u X , XI) 9
Em(X2
9
XI n X2>
and Em(Xl u X , , X I ) E"(X,, X , n X , ) . Proof To prove B, let X , = A and X , = X - U. Then Int X , u Int X , = Int A u X - B = X . Conversely, if B holds, making the same substitutions we see that D = X - Int X , c Int X , = Tnt A .
Definition 19.4 Let E be the atcegory of pairs ( X , A ) E CS2 with the AHEP. Let N = E n CS* be the category of well-pointed spaces. Theorem 19.5 On G, Emand Emare unreduced homology and cohomology theories.
-
Proof Iff: ( X , A ) ( Y , B ) define X u CA -+ Y u CB by f ( x ) = f ( x ) and f(a, t ) = ( f ( a ) ,t ) . y i s base point preserving. Furthermore, iff g in G, f- in CS*. Since X u CA N X / A , Em(X, u X , , X , ) g Em(X,, X, n),'A and Em(Xl u X , , X , ) r E m ( X 2 , X, n X,) with no hypothesis, for X , u X 2 / X 1 -+
19. Spectra[ Unreduced Homology and Cohomology Theories
185
= X,/X, n X,. (By Exercise 19, Section 14, XI is closed.) Exactness follows immediately from the homeomorphism
x+ u C*A+ = X U CA and 18.9.
I
Corollary 19.6
Em(X)Z Em(X,*) 0 Em(*),
E m ( X )g Em(X,*) @ Em(*).
Proof The exact sequences -+
-.
*
Em(*)+ E,,,(X) -+ Em(X,*) + . * * t E"(X, *)t
+ Ern(*)+ E"(X)
split since there is a map px: X 4 * with px i = 1 * by Exercise 11, Section 11 and 11.11. I Proposition 19.7 If X E A', Em(X, *) g E m ( X ) and Em(X, *) z E m ( X ) . Hence Em(X)E Em(X)0 Em(*)
and
E m ( X )E E m ( X )@ E m ( * ) .
Proof Let X * = X u C*. Then Em(X, *) =Em(X*) and E m ( X , *) = E m ( X * ) .It is sufficient to show that if X E N', ( X , *) N (X*, 1). There is an
obvious map a : X * + X in eG*, and since X E N there is a retraction P : X x / - + X * . L e t y ( x ) = P ( x , l ) . T h e n y : X + K * i s i n C G * . Nowap: 1 ay. Consider H : X* x I+ X x Z defined by H(x, s) = ( x , s), H(t, s) = (*, s + t(1 - s)). Then P H : 1 ya and hence ( X * , 1) N ( X , *). The second statement follows by applying 19.6. I
-
Corollary 19.8 Em(*)g Em(So)z E-"(*).
I
Corollary 19.9 If X E N,
Proposition 19.10 I f X is a CW complex, A m ( X ;n) LZ fi,,(X; n) = 0 for m<0. Proof X A K ( q n) is a CW complex with all cells in dimensions n and larger, except for 0-cells. Hence n m + , , ( X K(T, ~ n ) ) = 0 for m < 0 and n > 1.
19. Spectral Unreduced Homology and Cohomology Theories
186
Thus R J X ; n ) = 0 for m < 0. Since Sm-"Xhas all cells in dimensions n - m and larger, except for 0-cells, [Sn-mX,K(n, n)] = 0 if m < 0 by 16.3 applied to the diagram
The Eilenberg-Steenrod axioms on a positively graded functor are the laws: (A), (B), and (Cj of 19.2 together with the dimension axiom:
(D) A functor satisfying these properties is called an ordinary homology or cohomology theory. They are characteristic properties and are often used as a starting point for making calculations with ordinary homology and cohomology. By 19.5, 19.9, and 19.10 ordinary spectral homology and cohomology satisfy the Eilenberg-Steenrod axioms on the category of CW pairs.
Exercises
1." Prove that if X * * *
* * .
4
+-
3
A
3
B, there are long exact sequences
&(A, B ) -+ Em(X,B ) -+ Em(X,A )
A Em-,(A,B )
-+ *
*.
d
Ern(A,B) t E"(X, B ) t E y X , A ) t E m - l ( A , B ) 4- f . . .
(Compare with Exercise 5, Section 10.) 2. Let En ( E n )be an unreduced homology (cohomology) theory. Define En(X) = En(X, *) (i?"(X) = E"(X, *)). Show that ,!?,,(I?") is a reduced homology (cohomology) theory with the modification that S X is replaced by C X in Axiom (B) and X u C*A is replaced by X u CA in Axiom (C).
3. Show that E n ( u X a )2 0 En(X,), and if X , are CW complexes and E is an 0-spectrum, E"(UX,) r n E " ( X , ) . The assumption that the X , are CW complexes may be dropped if we assume that En is a CW complex for each n (see footnote 15). 4. Let { U,} be an open cover of X such that for all LY, LY' there exists LY" with U , u U,. c U a - .Show that E n ( X )E lim En(U,). (26.22, 26.28)