MathOverflow

I am trying to understand an apparent contradiction in the literature about degree-zero component $M_0(T_1,S^2)$ of the space of free maps from the torus $S^1\times S^1$ to the sphere $S^2$.
In Herbert Federer's 1956 paper "A Study of Function Spaces by Spectral Sequences" (top of page 358), he states that $\pi_1((S^2)^{S^1\times S^1}),w)$ is non-abelian, where $w$ is a constant map. Hansen also appears to be aware of this result: in his...

To show that there is no entire function $f:\mathbb{C}\to\mathbb{C}$ such that $\text{Re}(f(z))\text{Re}(f(w))<0$ or $\text{Im}(f(z))\text{Im}(f(w))<0$ whenever $|z-w|=1$, one can use the fact that $\chi(\mathbb{R}^2)>4$ (see here).
Is there a complex analytic proof of this fact?...

Let $\mathbf a,\mathbf b\in\mathbb R^{n}$ be generic vectors, and $(s,t)\in\left[0,1\right]^2$. Define
$$A=|\mathbf a-\mathbf b|^{2},\qquad B(s)=|s\mathbf a+(1-s)\mathbf b|^{2},\qquad C=\sqrt{\|\mathbf{a}\|^2\|\mathbf{b}\|^2-(\mathbf{a}\cdot\mathbf{b})^2},$$...

I am looking for examples of additively idempotent semirings (which are always join semilattices) that do not have an underlying lattice structure, i.e. either meets do not exist or exist outside the semiring considered. It is certain that any such example...