# Questions tagged [torus]

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6 questions
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### Topology of connected subsets of the $3$-torus

Consider the $3$-torus $Y=T^3$, a subset $\Sigma\subset Y$, and $\Sigma^*=Y\setminus\overline\Sigma$. We assume both $\Sigma$ and $\Sigma^*$ to be open, connected, and smoothly bounded. I am ...
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### On some finiteness properties of cohomological algebras of complex tori

Denote by $A := \mathbb{C}[u,u^{-1}]$, $u = (u_1,...,u_n)$, the algebra of polynomial functions on the complex torus $(\mathbb{C} \setminus \{ 0 \})^n$, which we consider as a $\mathbb{C}[u]$-module. ...
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### How can one define “punctured torus” in Homotopy Type Theory? Is its fundamental group the free product of the integers with themselves?

Questions. Has the beautiful old idea, in part already known to Gauß, of a punctured torus surface (take, if you will, the classical set-theoretic definition as the meaning of the latter three words)...
29 views

### Low bound approximation of a Torus Knot length

Is there a formula for approximating (lower bound) the torus knot length ? The torus knot parameters are (p, q, R, r) where (p,q) are co-primes and R is major axis and r is minor axis of the torus.
Let $T^n$ denote the $n$-dimensional torus. Suppose there is an open subset $U\subset T^n$ not containing any nontrivial loop. Does this imply that the inclusion $U\hookrightarrow T^n$ is ...
### Let $V$ and $W$ be real representations of a torus $T$ s.t. $\dim V^H=\dim W^H$, $\forall H<T$. Show that $V\simeq W$
$V^H:=\{v\in V:hv=v,\,\forall h\in H\}$ is the fixed point set. One has that $$V=\bigoplus_jV(\chi_j)\oplus V^T,$$ with $\chi_j:T\to U(1)$ a character of some nontrivial irreducible subrepresentation ...