WWGD said:Hi, let p : E--->B be a covering map. Then we have a result that for every subgroup of ## \pi_1(B) ## we have an associated covering map. Now, going in sort-of the reverse direction, is there a way of figuring out what ## \pi_1(B) ## is, if we know a collection of covering maps for B; what information do we need for these to be able to reconstruct ## \pi_1 (B) ## up to isomorphism?
WWGD said:Hi lavinia, I assume the space whose fundamental group is ZxZ is the base of some covering. If so, is there any constraint for the top space, is the top space the universal covering space? I guess no, since the deck group would then be ZxZ. And if I looked it up correctly, if the covering is regular, then there is a homeomorphism ## E /( \frac {\mathbb Z}{2 \mathbb Z} ) \simeq B ##, where B is the base. But then we would have ## \pi_1( E/( \mathbb Z / 2\mathbb Z))=\mathbb Z \times \mathbb Z ## , which cannot be. I hope I understood your question correctly.
Well, no, I don't think so; I was trying to construct a sort-of converse to the result that : given a cover p:E-->X, for every subgroup of ##G:=\pi_1(X)## , there is an associated covering map for every subgroup of G. Now, if we did not know the fundamental group of ##X## , but we knew all the covering maps p: E-->X, could we figure out ## G:= \pi_1(X) ## ? It is difficult to associate a subgroup for every cover, unless , I think, you can show that some are covers of others. And there is the additional issue : is it possible for different groups with non-trivial subgroups to share the exact same subgroups?lavinia said:What I meant to say was that the space with fundamental group ZxZ covers another space and the covering is a 2 fold covering. Can you deduce the fundamental group of the base? I was asking if that was your question.
WWGD said:Well, no, I don't think so; I was trying to construct a sort-of converse to the result that : given a cover p:E-->X, for every subgroup of ##G:=\pi_1(X)## , there is an associated covering map for every subgroup of G. Now, if we did not know the fundamental group of ##X## , but we knew all the covering maps p: E-->X, could we figure out ## G:= \pi_1(X) ## ? It is difficult to associate a subgroup for every cover, unless , I think, you can show that some are covers of others. And there is the additional issue : is it possible for different groups with non-trivial subgroups to share the exact same subgroups?
The purpose of reconstructing a group from covering maps is to gain a better understanding of the underlying structure and relationships between elements in a group. By analyzing the covering maps and their corresponding group actions, we can reconstruct the original group and its properties.
Covering maps are mappings between two topological spaces that preserve the local structure of the spaces. They relate to groups through their group actions, which are actions of a group on a set that preserve the group structure. Covering maps can be used to study the group actions and ultimately reconstruct the group.
No, not every group can be reconstructed from its covering maps. Only certain groups, such as finite groups and discrete groups, can be reconstructed using covering maps.
The process of reconstructing a group from its covering maps involves identifying the group actions and their corresponding covering maps, determining the relationship between the maps, and then using this information to reconstruct the group and its properties.
Reconstructing a group from its covering maps has various applications in mathematics and physics. It can be used to study symmetry groups, topological groups, and other mathematical structures. In physics, it can be used to understand the symmetries of physical systems, such as in quantum mechanics and field theory.