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2009 September 29th - questions

1 Harold W

Two questions about the forgetful functor from smooth (C-infinity) vector bundles to smooth fiber bundles:

A) Suppose that $E\to B$ is a smooth vector bundle (E,B smooth manifolds). Does $E\to B$ as a smooth fiber bundle uniquely determine $E\to B$ as a vector bundle, up to isomorphism?

In other words, if $E_1\to B_1$, $E_2\to B_2$ are vector bundles which are isomorphic as smooth fiber bundles (via some fiber-preserving diffeomorphism, not necessarily linear on fibers), are they necessarily isomorphic as smooth vector bundles (via some fiber-preserving diffeomorphism, linear on fibers)?

B) Suppose that $E\to B$ is a smooth fiber bundle with fibers diffeomorphic to $\mathbb{R}^n$. Can it be given the linear structure of a vector bundle?

2 Critch

Every finitey generated abelian group can be made into a ring, by the structure theorem. What is an abelian group that cannot be made into a ring?

3 Jason F

A) Does ZF (without the axiom of choice) imply that a well-ordering of $\mathbb{R}$ exists? Why/why not?

B) If we assume a well ordering exists on $\mathbb{R}$ what other cardinalities can can be proven well-orderable?

4 Darsh

The set S of (isomorphism classes of) compact connected 3-manifolds, with the operation of connected sum, is a monoid. The connected sum $X \# Y$ is the result of removing an 3-ball from each of $X$ and $Y$ and gluing them along the boundary of this ball (this operation is well-defined up to homeomorphism).

What is a minimal set of generators for this monoid? (In the 2D case, the torus and the projective plane suffice.)

5 Scott M

Is there a good version of Artin-Wedderburn for semisimple algebra objects? (click for elaboration.)

6 Zack

This question is about "approximating" an open set $U$ by an open set $V$.

Say $X$ is a metric space, $U,V\subset X$ connected opens such that $\overline{U\Delta V} \subset B(r,\partial U)$ (the closure symmetric difference of $U$ and $V$ is contained in a radius-r neighborhood of the boundary of $U$ ... basically, U and V only differ near the boundary of U).

Does there exist a continuous surjection $f: U\to V$ which is "r-close to the identity", meaning that $\sup(d(f,id)) < r$. ?

7 Pablo

The cardinality of {A, B, C,...., Z} can be written in base 10 as 26, in base 2 as 11010.  Expressing natural numbers N,M in a certain base has the advantage that there are simple rules for how to express the numbers NM and N+M in the same base.  Is there a way to express number using unique prime factorization that still allows you to do arithmetic?  Or more generally is there any naming scheme for the natural numbers that doesn't use any sort of base that still allows you to do arithmetic?

For example, one might try something like saying 1 = 1, 2 = (1), 2*3 = (1)(1), 2*5 = (1)()(1), etc.  But its not clear how to do arithmetic with this notation.