Shicheng Wang - Some connections between topology and number theory

Abstract

We will discuss some connections between topology and number theory inspired by the studies of mapping degrees and achirality of manifolds.

Here all mfd are closed and oriented.

Content

We define $D(M,N)$ as the set of all possible degrees of functions between $M\to N$. $D(M):=D(M,M)$.

Examples:

• $D(S^1)=\mathbb{Z}$
• $D(T_{g=2}^{d=2})=\{\pm 1,0\}$
• $D(\mathbb{C}{P}^n)=\{k^n:k\in \mathbb{Z}\}$

Next we want to study $3$-spaces. There are $8$ geometries in $\dim 3$: $S^3,E^3,S^2\times E,{\mathbb{H}}^2\times E^{\prime },Nil,Sol,\widetilde{PS{L}_2\mathbb{R}},{\mathbb{H}}^3$. For a 3D space $N^3$ it can be decomposed in a finite sum of geometries or almost geometries.

Theorems

• When $|D(M)|=\infty$ then $D(M)$ is computable.
• $|D(M)|=\infty ⟺$ either $N$ prime $N$ supports onf the the first $6$ geometries

Questions

• Can every set $A$ which includes $0$ be realized by $D(M,N)$? ⇒ No, by countability argument
• What is a concrete non-realizable set $A$?
• What number-theoretical properties does $A$ have?

We define the density by $\Delta (A)={\lim }_{n\to \infty }\frac{|A_n|}{2n}\in [0,1],\Delta (M)=\Delta (D(M))$ where $A_n=A\cup [-n,n]$ if the limit exists.

• Does the density exist? Yes! For a $3$-mfd the density is always rational. moreover the densities are dense in $[0,1]$