Content
Reconstruction problem: We have a structure group (like the Homeomorphism group. Can we reconstruct the original space from the group).
Current Results
Whittaker 1963: For compact manifolds . van Mill ‘84 showed something similar.
Takens/Filipkiewicz 1981: Banyaga 1990s generalized the theorem based on deep results of Oxtoby - Ulam, Calabi, Fathi. The above holds
Logical Rigidity
The theory of a group is the set of all finite first-order group theoretic sentences which are true about a group. (just a set of symbols, which does not give information about specific group elements but rather exists/for all statements).
Logical Recostruction Problem: Can we reconstruct a group from its theory?
How can we define allowed symbols for a theory?
- logical symols, for all, exists, not and, or, …
- non-logical symb
- variables
- signature: Constants, Functions, Rel, arity.
Some Theorem
Tarskis Problems:
- Is the Theory of free groups (even of different size) the same? Yes!
- Is the theory of free groups decidable? I.e. for any first-order statement, can a computer find out whether it is true or false?
There is something called logical rigidity which has do do with the above.
Invariants on Manifolds
The author found out that the manifold is determined by its theory.