Wedge axiom of a homology theory like functor
$begingroup$
Let $h_ncolon CW_*to Ab (nin mathbb{Z})$ be a covariant homotopy invariant functor that sends cofibre sequences to exact sequences and is equipped with natural suspension isomorphisms.
Then the following two statements are equivalent:
For every countable collection $(X_i)_{iin I}subseteq CW_*$ the natural morphism $bigoplus_i h_*(X_i)to h_*(bigvee_i X_i)$ is an isomorphism.
If $Y=colim_k(Y_0overset{cl.incl.}{hookrightarrow}Y_1hookrightarrowcdots)$, then the natural morphism $colim_k h_*(Y_k)to h_*(Y)$ is an Isomorphism.
I tried to show this, and had some ideas, but I still lack of understanding for a detailed proof. In particular, it is not clear to me how the suspension isomorphisms and the cofibre condition have to be used.
My considerations are as follows:
$2.implies 1.$
Assume 2. is true. Let $(X_i)_i$ be any countable collection of pointed CW-complexes. We may assume, that the index set are the natural numbers.
The most obvious thing that comes to my mind is to let $Y_k:=X_0vee cdots vee X_k$.
Then let $Y=colim(Y_0hookrightarrow Y_1hookrightarrow cdots)$.
(one issue here is, that I don't know how to replace the inclusions by closed ones, so I neglect this for a moment)
Now by 2. we know that the natural morphism $colim_k h_*(Y_k)to h_*(Y)$ is an isomorphism.
My intuition tells me that $h_*(X_0veecdots vee X_k)cong h_*(X_0)oplus cdots oplus h_*(X_k)$ should be true under the above assumptions, and that one has $colim_k (h_*(X_0)opluscdotsoplus h*(X_k))cong bigoplus_i h_*(X_i)$.
$1.implies 2.$
Assume 1. is true. Let $Y=colim(Y_0hookrightarrow Y_1hookrightarrowcdots)$, where the $hookrightarrow$ are closed inclusions.
If one could make the $Y_k$ become disjoint, the relations between them would disappear and the colimit would become a coproduct.
Hence one could replace $Y$ by the reduced mapping telescope $bigcup_{kgeq 0} [k,k+1]times Y_k$ and show that this doesn't affect the value of $h_*$.
By one we would have $bigoplus_k h_*(Y_k)cong colim_k h_*([k,k+1]times Y_k)cong colim_k h_*(Y_k)$.
-Is there a way to make this argument work?
-How do the suspension isomorphisms and the cofibre condition join the argument?
algebraic-topology homology-cohomology
asked Jan 7 at 9:05
sdigrsdigr
163
$endgroup$
add a comment |
$begingroup$
Let $h_ncolon CW_*to Ab (nin mathbb{Z})$ be a covariant homotopy invariant functor that sends cofibre sequences to exact sequences and is equipped with natural suspension isomorphisms.
Then the following two statements are equivalent:
For every countable collection $(X_i)_{iin I}subseteq CW_*$ the natural morphism $bigoplus_i h_*(X_i)to h_*(bigvee_i X_i)$ is an isomorphism.
If $Y=colim_k(Y_0overset{cl.incl.}{hookrightarrow}Y_1hookrightarrowcdots)$, then the natural morphism $colim_k h_*(Y_k)to h_*(Y)$ is an Isomorphism.
I tried to show this, and had some ideas, but I still lack of understanding for a detailed proof. In particular, it is not clear to me how the suspension isomorphisms and the cofibre condition have to be used.
My considerations are as follows:
$2.implies 1.$
Assume 2. is true. Let $(X_i)_i$ be any countable collection of pointed CW-complexes. We may assume, that the index set are the natural numbers.
The most obvious thing that comes to my mind is to let $Y_k:=X_0vee cdots vee X_k$.
Then let $Y=colim(Y_0hookrightarrow Y_1hookrightarrow cdots)$.
(one issue here is, that I don't know how to replace the inclusions by closed ones, so I neglect this for a moment)
Now by 2. we know that the natural morphism $colim_k h_*(Y_k)to h_*(Y)$ is an isomorphism.
My intuition tells me that $h_*(X_0veecdots vee X_k)cong h_*(X_0)oplus cdots oplus h_*(X_k)$ should be true under the above assumptions, and that one has $colim_k (h_*(X_0)opluscdotsoplus h*(X_k))cong bigoplus_i h_*(X_i)$.
$1.implies 2.$
Assume 1. is true. Let $Y=colim(Y_0hookrightarrow Y_1hookrightarrowcdots)$, where the $hookrightarrow$ are closed inclusions.
If one could make the $Y_k$ become disjoint, the relations between them would disappear and the colimit would become a coproduct.
Hence one could replace $Y$ by the reduced mapping telescope $bigcup_{kgeq 0} [k,k+1]times Y_k$ and show that this doesn't affect the value of $h_*$.
By one we would have $bigoplus_k h_*(Y_k)cong colim_k h_*([k,k+1]times Y_k)cong colim_k h_*(Y_k)$.
-Is there a way to make this argument work?
-How do the suspension isomorphisms and the cofibre condition join the argument?
algebraic-topology homology-cohomology
asked Jan 7 at 9:05
sdigrsdigr
163
$endgroup$
$begingroup$
You consider a reduced homology theory on $CW_*$. It is well-known that then $h_*(X vee Y) approx h_*(X) oplus h_*(Y)$ which generalizes to finite wedges. Moreover, the inclusion $X_1 vee dots vee X_k hookrightarrow X_1 vee dots vee X_{k+1}$ is closed.
$endgroup$
– Paul Frost
Jan 7 at 9:29
$begingroup$
1. $Rightarrow$ 2. is not all trivial. You can find a proof in Switzer, Robert M. Algebraic Topology - Homotopy and Homology. Springer, 2017. See Proposition 7.53 which has a two-pages-proof (plus the material proved previously).
$endgroup$
– Paul Frost
Jan 7 at 13:28
add a comment |
$begingroup$
Let $h_ncolon CW_*to Ab (nin mathbb{Z})$ be a covariant homotopy invariant functor that sends cofibre sequences to exact sequences and is equipped with natural suspension isomorphisms.
Then the following two statements are equivalent:
For every countable collection $(X_i)_{iin I}subseteq CW_*$ the natural morphism $bigoplus_i h_*(X_i)to h_*(bigvee_i X_i)$ is an isomorphism.
If $Y=colim_k(Y_0overset{cl.incl.}{hookrightarrow}Y_1hookrightarrowcdots)$, then the natural morphism $colim_k h_*(Y_k)to h_*(Y)$ is an Isomorphism.
I tried to show this, and had some ideas, but I still lack of understanding for a detailed proof. In particular, it is not clear to me how the suspension isomorphisms and the cofibre condition have to be used.
My considerations are as follows:
$2.implies 1.$
Assume 2. is true. Let $(X_i)_i$ be any countable collection of pointed CW-complexes. We may assume, that the index set are the natural numbers.
The most obvious thing that comes to my mind is to let $Y_k:=X_0vee cdots vee X_k$.
Then let $Y=colim(Y_0hookrightarrow Y_1hookrightarrow cdots)$.
(one issue here is, that I don't know how to replace the inclusions by closed ones, so I neglect this for a moment)
Now by 2. we know that the natural morphism $colim_k h_*(Y_k)to h_*(Y)$ is an isomorphism.
My intuition tells me that $h_*(X_0veecdots vee X_k)cong h_*(X_0)oplus cdots oplus h_*(X_k)$ should be true under the above assumptions, and that one has $colim_k (h_*(X_0)opluscdotsoplus h*(X_k))cong bigoplus_i h_*(X_i)$.
$1.implies 2.$
Assume 1. is true. Let $Y=colim(Y_0hookrightarrow Y_1hookrightarrowcdots)$, where the $hookrightarrow$ are closed inclusions.
If one could make the $Y_k$ become disjoint, the relations between them would disappear and the colimit would become a coproduct.
Hence one could replace $Y$ by the reduced mapping telescope $bigcup_{kgeq 0} [k,k+1]times Y_k$ and show that this doesn't affect the value of $h_*$.
By one we would have $bigoplus_k h_*(Y_k)cong colim_k h_*([k,k+1]times Y_k)cong colim_k h_*(Y_k)$.
-Is there a way to make this argument work?
-How do the suspension isomorphisms and the cofibre condition join the argument?
algebraic-topology homology-cohomology
asked Jan 7 at 9:05
sdigrsdigr
163
$endgroup$
Let $h_ncolon CW_*to Ab (nin mathbb{Z})$ be a covariant homotopy invariant functor that sends cofibre sequences to exact sequences and is equipped with natural suspension isomorphisms.
Then the following two statements are equivalent:
For every countable collection $(X_i)_{iin I}subseteq CW_*$ the natural morphism $bigoplus_i h_*(X_i)to h_*(bigvee_i X_i)$ is an isomorphism.
If $Y=colim_k(Y_0overset{cl.incl.}{hookrightarrow}Y_1hookrightarrowcdots)$, then the natural morphism $colim_k h_*(Y_k)to h_*(Y)$ is an Isomorphism.
I tried to show this, and had some ideas, but I still lack of understanding for a detailed proof. In particular, it is not clear to me how the suspension isomorphisms and the cofibre condition have to be used.
My considerations are as follows:
$2.implies 1.$
Assume 2. is true. Let $(X_i)_i$ be any countable collection of pointed CW-complexes. We may assume, that the index set are the natural numbers.
The most obvious thing that comes to my mind is to let $Y_k:=X_0vee cdots vee X_k$.
Then let $Y=colim(Y_0hookrightarrow Y_1hookrightarrow cdots)$.
(one issue here is, that I don't know how to replace the inclusions by closed ones, so I neglect this for a moment)
Now by 2. we know that the natural morphism $colim_k h_*(Y_k)to h_*(Y)$ is an isomorphism.
My intuition tells me that $h_*(X_0veecdots vee X_k)cong h_*(X_0)oplus cdots oplus h_*(X_k)$ should be true under the above assumptions, and that one has $colim_k (h_*(X_0)opluscdotsoplus h*(X_k))cong bigoplus_i h_*(X_i)$.
$1.implies 2.$
Assume 1. is true. Let $Y=colim(Y_0hookrightarrow Y_1hookrightarrowcdots)$, where the $hookrightarrow$ are closed inclusions.
If one could make the $Y_k$ become disjoint, the relations between them would disappear and the colimit would become a coproduct.
Hence one could replace $Y$ by the reduced mapping telescope $bigcup_{kgeq 0} [k,k+1]times Y_k$ and show that this doesn't affect the value of $h_*$.
By one we would have $bigoplus_k h_*(Y_k)cong colim_k h_*([k,k+1]times Y_k)cong colim_k h_*(Y_k)$.
-Is there a way to make this argument work?
-How do the suspension isomorphisms and the cofibre condition join the argument?
algebraic-topology homology-cohomology
algebraic-topology homology-cohomology
asked Jan 7 at 9:05
sdigrsdigr
163
asked Jan 7 at 9:05
sdigrsdigr
163
asked Jan 7 at 9:05
sdigrsdigr
163
asked Jan 7 at 9:05
sdigrsdigr
163
asked Jan 7 at 9:05
sdigrsdigr
163
163
$begingroup$
You consider a reduced homology theory on $CW_*$. It is well-known that then $h_*(X vee Y) approx h_*(X) oplus h_*(Y)$ which generalizes to finite wedges. Moreover, the inclusion $X_1 vee dots vee X_k hookrightarrow X_1 vee dots vee X_{k+1}$ is closed.
$endgroup$
– Paul Frost
Jan 7 at 9:29
$begingroup$
1. $Rightarrow$ 2. is not all trivial. You can find a proof in Switzer, Robert M. Algebraic Topology - Homotopy and Homology. Springer, 2017. See Proposition 7.53 which has a two-pages-proof (plus the material proved previously).
$endgroup$
– Paul Frost
Jan 7 at 13:28
add a comment |
$begingroup$
You consider a reduced homology theory on $CW_*$. It is well-known that then $h_*(X vee Y) approx h_*(X) oplus h_*(Y)$ which generalizes to finite wedges. Moreover, the inclusion $X_1 vee dots vee X_k hookrightarrow X_1 vee dots vee X_{k+1}$ is closed.
$endgroup$
– Paul Frost
Jan 7 at 9:29
$begingroup$
1. $Rightarrow$ 2. is not all trivial. You can find a proof in Switzer, Robert M. Algebraic Topology - Homotopy and Homology. Springer, 2017. See Proposition 7.53 which has a two-pages-proof (plus the material proved previously).
$endgroup$
– Paul Frost
Jan 7 at 13:28
$begingroup$
You consider a reduced homology theory on $CW_*$. It is well-known that then $h_*(X vee Y) approx h_*(X) oplus h_*(Y)$ which generalizes to finite wedges. Moreover, the inclusion $X_1 vee dots vee X_k hookrightarrow X_1 vee dots vee X_{k+1}$ is closed.
$endgroup$
– Paul Frost
Jan 7 at 9:29
$begingroup$
You consider a reduced homology theory on $CW_*$. It is well-known that then $h_*(X vee Y) approx h_*(X) oplus h_*(Y)$ which generalizes to finite wedges. Moreover, the inclusion $X_1 vee dots vee X_k hookrightarrow X_1 vee dots vee X_{k+1}$ is closed.
$endgroup$
– Paul Frost
Jan 7 at 9:29
$begingroup$
1. $Rightarrow$ 2. is not all trivial. You can find a proof in Switzer, Robert M. Algebraic Topology - Homotopy and Homology. Springer, 2017. See Proposition 7.53 which has a two-pages-proof (plus the material proved previously).
$endgroup$
– Paul Frost
Jan 7 at 13:28
$begingroup$
1. $Rightarrow$ 2. is not all trivial. You can find a proof in Switzer, Robert M. Algebraic Topology - Homotopy and Homology. Springer, 2017. See Proposition 7.53 which has a two-pages-proof (plus the material proved previously).
$endgroup$
– Paul Frost
Jan 7 at 13:28
add a comment |
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$begingroup$
You consider a reduced homology theory on $CW_*$. It is well-known that then $h_*(X vee Y) approx h_*(X) oplus h_*(Y)$ which generalizes to finite wedges. Moreover, the inclusion $X_1 vee dots vee X_k hookrightarrow X_1 vee dots vee X_{k+1}$ is closed.
$endgroup$
– Paul Frost
Jan 7 at 9:29
$begingroup$
1. $Rightarrow$ 2. is not all trivial. You can find a proof in Switzer, Robert M. Algebraic Topology - Homotopy and Homology. Springer, 2017. See Proposition 7.53 which has a two-pages-proof (plus the material proved previously).
$endgroup$
– Paul Frost
Jan 7 at 13:28