Show that every compact hypersurface in $\mathbb{R}^n$ is orientable. HINT: Jordan-Brouwer Separation Theorem. This is an exercise from Guillemin and Pollack. So hypersurface means smooth hypersurface. Jordan's Theorem says that if $S$ is some hypersurface in $\mathbb{R}^n$, then $\mathbb{R}...

Let $A$ be a $d$-dimensional algebra over a field $K$. One can naturally assign to $A$ the linear algebraic $K$-group $\mathbf{GL_1}(A)$ that represents the functor $B \mapsto (A \otimes_K B)^\times$ for $K$-algebras $B$. My question is: Is the functor $A \mapsto \mathbf{GL_1}(A)$ from finite...

If $\{f_n\}$ is a sequence of bounded measure functions on $[a,b]$ and $f_n \to f$ uniformly, then $f$ is integrable. This is actually proved in the book I am working through, but I came up with a proof that looks quite different. Naturally, I was hoping someone could critique my proof and...

Let $M$ be a compact riemannian manifold with boundary $\partial M$. We have that $\partial M$ is also compact and I was able to show that there is some $a>0$ such that the map $F:[0,a]\times \partial M \to M$ given by $F(r,p)= \exp_p(r\nu(p))$ is a diffeomorphism onto its image, say $U$ (here $\...