Homotopy fibre of pushout
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Let $(f_U,f_A,f_V):(Uleftarrow Arightarrow V)to (U'leftarrow A'rightarrow V')$ be a morphism of diagrams of topological spaces and let $f:Usqcup_A Vto U'sqcup_{A'} V'$ be the resulting morphisms for the pushouts.
Suppose I know the (weak) homotopy type of the homotopy fibres of $f_U$, $f_A$ and $f_V$. Is there any chance to say something about the homotopy fibre of $f$?
To be more precise: In my special situation, I know that $f_U$, $f_A$ and $f_V$ are all coverings and I would be happy if I could conclude that $pi_2(mathrm{hofib}(f))=0$.
homotopy-theory covering-spaces fiber-bundles
asked Nov 16 at 13:27
FKranhold
1787
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Let $(f_U,f_A,f_V):(Uleftarrow Arightarrow V)to (U'leftarrow A'rightarrow V')$ be a morphism of diagrams of topological spaces and let $f:Usqcup_A Vto U'sqcup_{A'} V'$ be the resulting morphisms for the pushouts.
Suppose I know the (weak) homotopy type of the homotopy fibres of $f_U$, $f_A$ and $f_V$. Is there any chance to say something about the homotopy fibre of $f$?
To be more precise: In my special situation, I know that $f_U$, $f_A$ and $f_V$ are all coverings and I would be happy if I could conclude that $pi_2(mathrm{hofib}(f))=0$.
homotopy-theory covering-spaces fiber-bundles
asked Nov 16 at 13:27
FKranhold
1787
I can't think of anything direct, unless you have more information available about the diagram map. I think your best bet will be to study the cofibers of the map. In this case $C_f$ is the homotopy pushout of $(C_{f_U}leftarrow C_{f_A}rightarrow C_{f_V})$, and you may be able to get a good grasp of its cohomology. Then use this to show that $f$ is sufficiently connected so as to be able to conclude that $pi_2F_f=0$.
– Tyrone
Nov 16 at 13:37
I doubt it. Consider $U'=A'=V' = {*}$ (singleton) and $U = V$ disks, glued along $A = S^1$. Then the homotopy fiber of $f$ is $S^2$. These are not covering maps, but they are fibrations...
– Najib Idrissi
Nov 16 at 13:38
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $(f_U,f_A,f_V):(Uleftarrow Arightarrow V)to (U'leftarrow A'rightarrow V')$ be a morphism of diagrams of topological spaces and let $f:Usqcup_A Vto U'sqcup_{A'} V'$ be the resulting morphisms for the pushouts.
Suppose I know the (weak) homotopy type of the homotopy fibres of $f_U$, $f_A$ and $f_V$. Is there any chance to say something about the homotopy fibre of $f$?
To be more precise: In my special situation, I know that $f_U$, $f_A$ and $f_V$ are all coverings and I would be happy if I could conclude that $pi_2(mathrm{hofib}(f))=0$.
homotopy-theory covering-spaces fiber-bundles
asked Nov 16 at 13:27
FKranhold
1787
Let $(f_U,f_A,f_V):(Uleftarrow Arightarrow V)to (U'leftarrow A'rightarrow V')$ be a morphism of diagrams of topological spaces and let $f:Usqcup_A Vto U'sqcup_{A'} V'$ be the resulting morphisms for the pushouts.
Suppose I know the (weak) homotopy type of the homotopy fibres of $f_U$, $f_A$ and $f_V$. Is there any chance to say something about the homotopy fibre of $f$?
To be more precise: In my special situation, I know that $f_U$, $f_A$ and $f_V$ are all coverings and I would be happy if I could conclude that $pi_2(mathrm{hofib}(f))=0$.
homotopy-theory covering-spaces fiber-bundles
homotopy-theory covering-spaces fiber-bundles
asked Nov 16 at 13:27
FKranhold
1787
asked Nov 16 at 13:27
FKranhold
1787
asked Nov 16 at 13:27
FKranhold
1787
asked Nov 16 at 13:27
FKranhold
1787
asked Nov 16 at 13:27
FKranhold
1787
1787
I can't think of anything direct, unless you have more information available about the diagram map. I think your best bet will be to study the cofibers of the map. In this case $C_f$ is the homotopy pushout of $(C_{f_U}leftarrow C_{f_A}rightarrow C_{f_V})$, and you may be able to get a good grasp of its cohomology. Then use this to show that $f$ is sufficiently connected so as to be able to conclude that $pi_2F_f=0$.
– Tyrone
Nov 16 at 13:37
I doubt it. Consider $U'=A'=V' = {*}$ (singleton) and $U = V$ disks, glued along $A = S^1$. Then the homotopy fiber of $f$ is $S^2$. These are not covering maps, but they are fibrations...
– Najib Idrissi
Nov 16 at 13:38
add a comment |
I can't think of anything direct, unless you have more information available about the diagram map. I think your best bet will be to study the cofibers of the map. In this case $C_f$ is the homotopy pushout of $(C_{f_U}leftarrow C_{f_A}rightarrow C_{f_V})$, and you may be able to get a good grasp of its cohomology. Then use this to show that $f$ is sufficiently connected so as to be able to conclude that $pi_2F_f=0$.
– Tyrone
Nov 16 at 13:37
I doubt it. Consider $U'=A'=V' = {*}$ (singleton) and $U = V$ disks, glued along $A = S^1$. Then the homotopy fiber of $f$ is $S^2$. These are not covering maps, but they are fibrations...
– Najib Idrissi
Nov 16 at 13:38
I can't think of anything direct, unless you have more information available about the diagram map. I think your best bet will be to study the cofibers of the map. In this case $C_f$ is the homotopy pushout of $(C_{f_U}leftarrow C_{f_A}rightarrow C_{f_V})$, and you may be able to get a good grasp of its cohomology. Then use this to show that $f$ is sufficiently connected so as to be able to conclude that $pi_2F_f=0$.
– Tyrone
Nov 16 at 13:37
I can't think of anything direct, unless you have more information available about the diagram map. I think your best bet will be to study the cofibers of the map. In this case $C_f$ is the homotopy pushout of $(C_{f_U}leftarrow C_{f_A}rightarrow C_{f_V})$, and you may be able to get a good grasp of its cohomology. Then use this to show that $f$ is sufficiently connected so as to be able to conclude that $pi_2F_f=0$.
– Tyrone
Nov 16 at 13:37
I doubt it. Consider $U'=A'=V' = {*}$ (singleton) and $U = V$ disks, glued along $A = S^1$. Then the homotopy fiber of $f$ is $S^2$. These are not covering maps, but they are fibrations...
– Najib Idrissi
Nov 16 at 13:38
I doubt it. Consider $U'=A'=V' = {*}$ (singleton) and $U = V$ disks, glued along $A = S^1$. Then the homotopy fiber of $f$ is $S^2$. These are not covering maps, but they are fibrations...
– Najib Idrissi
Nov 16 at 13:38
add a comment |
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I can't think of anything direct, unless you have more information available about the diagram map. I think your best bet will be to study the cofibers of the map. In this case $C_f$ is the homotopy pushout of $(C_{f_U}leftarrow C_{f_A}rightarrow C_{f_V})$, and you may be able to get a good grasp of its cohomology. Then use this to show that $f$ is sufficiently connected so as to be able to conclude that $pi_2F_f=0$.
– Tyrone
Nov 16 at 13:37
I doubt it. Consider $U'=A'=V' = {*}$ (singleton) and $U = V$ disks, glued along $A = S^1$. Then the homotopy fiber of $f$ is $S^2$. These are not covering maps, but they are fibrations...
– Najib Idrissi
Nov 16 at 13:38