Mixing
Homology of the complex of exact sequences of homotopy groups
$begingroup$
By naturality of the connecting homomorphism, relative homotopy $pi_{bullet}$ may be regarded as a functor from the category of pointed pairs of topological spaces to the category of exact sequences of pointed sets (actually consisting of groups and homomorphisms from some level). Thus it transforms an exact sequence of pointed pairs into a chain complex of exact sequences. I am curious if this complex is exact. If not, what is its homology?
Special case which I suspect to be particularly interesting is that of a pointed triple (X,A,B). Then we have an exact sequence of pairs
$$ (*,*) to (A,B) to (X,A) to left( frac{X}{A}, * right) to (*,*), $$
where $*$ is a singleton. This yields a complex of exact sequences
$$ 0 to pi_{bullet}(A,B) to pi_{bullet}(X,A) to pi_{bullet} left( frac{X}{A} right) to 0. $$
algebraic-topology homology-cohomology homotopy-theory exact-sequence functors
asked Jan 27 at 18:52
BlazejBlazej
1,632620
$endgroup$
add a comment |
$begingroup$
By naturality of the connecting homomorphism, relative homotopy $pi_{bullet}$ may be regarded as a functor from the category of pointed pairs of topological spaces to the category of exact sequences of pointed sets (actually consisting of groups and homomorphisms from some level). Thus it transforms an exact sequence of pointed pairs into a chain complex of exact sequences. I am curious if this complex is exact. If not, what is its homology?
Special case which I suspect to be particularly interesting is that of a pointed triple (X,A,B). Then we have an exact sequence of pairs
$$ (*,*) to (A,B) to (X,A) to left( frac{X}{A}, * right) to (*,*), $$
where $*$ is a singleton. This yields a complex of exact sequences
$$ 0 to pi_{bullet}(A,B) to pi_{bullet}(X,A) to pi_{bullet} left( frac{X}{A} right) to 0. $$
algebraic-topology homology-cohomology homotopy-theory exact-sequence functors
asked Jan 27 at 18:52
BlazejBlazej
1,632620
$endgroup$
$begingroup$
What you get from a triple is a long exact sequence, from a pair too, and from maps of pairs you get morphisms. But you don't get complexes, since your long exact sequences are... exact.
$endgroup$
– Pedro Tamaroff♦
Jan 31 at 10:30
$begingroup$
You should define what you mean by an exact sequence of pointed pairs. Which concept of a "complex" do you use? You certainly have a short sequence whose terms are long exact sequences, but this short sequence is in general not exact.
$endgroup$
– Paul Frost
Jan 31 at 11:17
$begingroup$
Kernel of a map of pointed pairs is defined as the preimage of the basepoint. Exactness means that kernel coincides with the image of the previous map. I do have a short sequence whose terms are long exact sequences and I ask precisely when this sequence is exact. I suspect it is not in general, so it makes sense to ask about its homology (at least starting from some degree, where we actually have abelian groups only).
$endgroup$
– Blazej
Jan 31 at 11:45
$begingroup$
My schematic writing of whole exact sequence as $pi_{bullet}(cdot, cdot)$ means that in fact there is a sequence extending indefinitely in vertical directions. All vertical arrows are exact and in every row we have a sequence of the form $0 to G_1 to G_2 to G_3 to 0$. I don't think this sequence is exact, so it could be interesting to describe its failure to be exact.
$endgroup$
– Blazej
Jan 31 at 11:59
add a comment |
$begingroup$
By naturality of the connecting homomorphism, relative homotopy $pi_{bullet}$ may be regarded as a functor from the category of pointed pairs of topological spaces to the category of exact sequences of pointed sets (actually consisting of groups and homomorphisms from some level). Thus it transforms an exact sequence of pointed pairs into a chain complex of exact sequences. I am curious if this complex is exact. If not, what is its homology?
Special case which I suspect to be particularly interesting is that of a pointed triple (X,A,B). Then we have an exact sequence of pairs
$$ (*,*) to (A,B) to (X,A) to left( frac{X}{A}, * right) to (*,*), $$
where $*$ is a singleton. This yields a complex of exact sequences
$$ 0 to pi_{bullet}(A,B) to pi_{bullet}(X,A) to pi_{bullet} left( frac{X}{A} right) to 0. $$
algebraic-topology homology-cohomology homotopy-theory exact-sequence functors
asked Jan 27 at 18:52
BlazejBlazej
1,632620
$endgroup$
By naturality of the connecting homomorphism, relative homotopy $pi_{bullet}$ may be regarded as a functor from the category of pointed pairs of topological spaces to the category of exact sequences of pointed sets (actually consisting of groups and homomorphisms from some level). Thus it transforms an exact sequence of pointed pairs into a chain complex of exact sequences. I am curious if this complex is exact. If not, what is its homology?
Special case which I suspect to be particularly interesting is that of a pointed triple (X,A,B). Then we have an exact sequence of pairs
$$ (*,*) to (A,B) to (X,A) to left( frac{X}{A}, * right) to (*,*), $$
where $*$ is a singleton. This yields a complex of exact sequences
$$ 0 to pi_{bullet}(A,B) to pi_{bullet}(X,A) to pi_{bullet} left( frac{X}{A} right) to 0. $$
algebraic-topology homology-cohomology homotopy-theory exact-sequence functors
algebraic-topology homology-cohomology homotopy-theory exact-sequence functors
asked Jan 27 at 18:52
BlazejBlazej
1,632620
asked Jan 27 at 18:52
BlazejBlazej
1,632620
asked Jan 27 at 18:52
BlazejBlazej
1,632620
asked Jan 27 at 18:52
BlazejBlazej
1,632620
asked Jan 27 at 18:52
BlazejBlazej
1,632620
1,632620
$begingroup$
What you get from a triple is a long exact sequence, from a pair too, and from maps of pairs you get morphisms. But you don't get complexes, since your long exact sequences are... exact.
$endgroup$
– Pedro Tamaroff♦
Jan 31 at 10:30
$begingroup$
You should define what you mean by an exact sequence of pointed pairs. Which concept of a "complex" do you use? You certainly have a short sequence whose terms are long exact sequences, but this short sequence is in general not exact.
$endgroup$
– Paul Frost
Jan 31 at 11:17
$begingroup$
Kernel of a map of pointed pairs is defined as the preimage of the basepoint. Exactness means that kernel coincides with the image of the previous map. I do have a short sequence whose terms are long exact sequences and I ask precisely when this sequence is exact. I suspect it is not in general, so it makes sense to ask about its homology (at least starting from some degree, where we actually have abelian groups only).
$endgroup$
– Blazej
Jan 31 at 11:45
$begingroup$
My schematic writing of whole exact sequence as $pi_{bullet}(cdot, cdot)$ means that in fact there is a sequence extending indefinitely in vertical directions. All vertical arrows are exact and in every row we have a sequence of the form $0 to G_1 to G_2 to G_3 to 0$. I don't think this sequence is exact, so it could be interesting to describe its failure to be exact.
$endgroup$
– Blazej
Jan 31 at 11:59
add a comment |
$begingroup$
What you get from a triple is a long exact sequence, from a pair too, and from maps of pairs you get morphisms. But you don't get complexes, since your long exact sequences are... exact.
$endgroup$
– Pedro Tamaroff♦
Jan 31 at 10:30
$begingroup$
You should define what you mean by an exact sequence of pointed pairs. Which concept of a "complex" do you use? You certainly have a short sequence whose terms are long exact sequences, but this short sequence is in general not exact.
$endgroup$
– Paul Frost
Jan 31 at 11:17
$begingroup$
Kernel of a map of pointed pairs is defined as the preimage of the basepoint. Exactness means that kernel coincides with the image of the previous map. I do have a short sequence whose terms are long exact sequences and I ask precisely when this sequence is exact. I suspect it is not in general, so it makes sense to ask about its homology (at least starting from some degree, where we actually have abelian groups only).
$endgroup$
– Blazej
Jan 31 at 11:45
$begingroup$
My schematic writing of whole exact sequence as $pi_{bullet}(cdot, cdot)$ means that in fact there is a sequence extending indefinitely in vertical directions. All vertical arrows are exact and in every row we have a sequence of the form $0 to G_1 to G_2 to G_3 to 0$. I don't think this sequence is exact, so it could be interesting to describe its failure to be exact.
$endgroup$
– Blazej
Jan 31 at 11:59
$begingroup$
What you get from a triple is a long exact sequence, from a pair too, and from maps of pairs you get morphisms. But you don't get complexes, since your long exact sequences are... exact.
$endgroup$
– Pedro Tamaroff♦
Jan 31 at 10:30
$begingroup$
What you get from a triple is a long exact sequence, from a pair too, and from maps of pairs you get morphisms. But you don't get complexes, since your long exact sequences are... exact.
$endgroup$
– Pedro Tamaroff♦
Jan 31 at 10:30
$begingroup$
You should define what you mean by an exact sequence of pointed pairs. Which concept of a "complex" do you use? You certainly have a short sequence whose terms are long exact sequences, but this short sequence is in general not exact.
$endgroup$
– Paul Frost
Jan 31 at 11:17
$begingroup$
You should define what you mean by an exact sequence of pointed pairs. Which concept of a "complex" do you use? You certainly have a short sequence whose terms are long exact sequences, but this short sequence is in general not exact.
$endgroup$
– Paul Frost
Jan 31 at 11:17
$begingroup$
Kernel of a map of pointed pairs is defined as the preimage of the basepoint. Exactness means that kernel coincides with the image of the previous map. I do have a short sequence whose terms are long exact sequences and I ask precisely when this sequence is exact. I suspect it is not in general, so it makes sense to ask about its homology (at least starting from some degree, where we actually have abelian groups only).
$endgroup$
– Blazej
Jan 31 at 11:45
$begingroup$
Kernel of a map of pointed pairs is defined as the preimage of the basepoint. Exactness means that kernel coincides with the image of the previous map. I do have a short sequence whose terms are long exact sequences and I ask precisely when this sequence is exact. I suspect it is not in general, so it makes sense to ask about its homology (at least starting from some degree, where we actually have abelian groups only).
$endgroup$
– Blazej
Jan 31 at 11:45
$begingroup$
My schematic writing of whole exact sequence as $pi_{bullet}(cdot, cdot)$ means that in fact there is a sequence extending indefinitely in vertical directions. All vertical arrows are exact and in every row we have a sequence of the form $0 to G_1 to G_2 to G_3 to 0$. I don't think this sequence is exact, so it could be interesting to describe its failure to be exact.
$endgroup$
– Blazej
Jan 31 at 11:59
$begingroup$
My schematic writing of whole exact sequence as $pi_{bullet}(cdot, cdot)$ means that in fact there is a sequence extending indefinitely in vertical directions. All vertical arrows are exact and in every row we have a sequence of the form $0 to G_1 to G_2 to G_3 to 0$. I don't think this sequence is exact, so it could be interesting to describe its failure to be exact.
$endgroup$
– Blazej
Jan 31 at 11:59
add a comment |
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$begingroup$
What you get from a triple is a long exact sequence, from a pair too, and from maps of pairs you get morphisms. But you don't get complexes, since your long exact sequences are... exact.
$endgroup$
– Pedro Tamaroff♦
Jan 31 at 10:30
$begingroup$
You should define what you mean by an exact sequence of pointed pairs. Which concept of a "complex" do you use? You certainly have a short sequence whose terms are long exact sequences, but this short sequence is in general not exact.
$endgroup$
– Paul Frost
Jan 31 at 11:17
$begingroup$
Kernel of a map of pointed pairs is defined as the preimage of the basepoint. Exactness means that kernel coincides with the image of the previous map. I do have a short sequence whose terms are long exact sequences and I ask precisely when this sequence is exact. I suspect it is not in general, so it makes sense to ask about its homology (at least starting from some degree, where we actually have abelian groups only).
$endgroup$
– Blazej
Jan 31 at 11:45
$begingroup$
My schematic writing of whole exact sequence as $pi_{bullet}(cdot, cdot)$ means that in fact there is a sequence extending indefinitely in vertical directions. All vertical arrows are exact and in every row we have a sequence of the form $0 to G_1 to G_2 to G_3 to 0$. I don't think this sequence is exact, so it could be interesting to describe its failure to be exact.
$endgroup$
– Blazej
Jan 31 at 11:59