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Register a new userThe Borsuk-Ulam property for homotopy classes of maps between the torus and the Klein bottle -- part 2
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Let $M$ be a topological space that admits a free involution $tau$, and let $N$ be a topological space. A homotopy class $beta in [ M,N ]$ is said to have the Borsuk-Ulam property with respect to $tau$ if for every representative map $f: M to N$ of $beta$, there exists a point $x in M$ such that $f(tau(x))= f(x)$. In this paper, we determine the homotopy class of maps from the $2$-torus $T^2$ to the Klein bottle $K^2$ that possess the Borsuk-Ulam property with respect to any free involution of $T^2$ for which the orbit space is $K^2$. Our results are given in terms of a certain family of homomorphisms involving the fundamental groups of $T^2$ and $K^2$. This completes the analysis of the Borsuk-Ulam problem for the case $M=T^2$ and $N=K^2$, and for any free involution $tau$ of $T^2$.
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Let $M$ be a topological space that admits a free involution $tau$, and let $N$ be a topological space. A homotopy class $beta in [ M,N ]$ is said to have {it the Borsuk-Ulam property with respect to $tau$} if for every representative map $f: M to N$ of $beta$, there exists a point $x in M$ such that $f(tau(x))= f(x)$. In this paper, we determine the homotopy classes of maps from the $2$-torus $T^2$ to the Klein bottle $K^2$ that possess the Borsuk-Ulam property with respect to a free involution $tau_1$ of $T^2$ for which the orbit space is $T^2$. Our results are given in terms of a certain family of homomorphisms involving the fundamental groups of $T^2$ and $K^2$.
Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the geometry of circle actions on Euclidean space, the structure of zeros of trigonometric polynomials, and theorems of Borsuk-Ulam type. We use the combinatorial and geometric structure of the convex hull of orbits of circle actions on Euclidean space to give geometric proofs of the homotopy type of metric thickenings of the circle. Homotopical connectivity bounds of thickenings of the sphere allow us to prove that a weighted average of function values of odd maps $S^n to mathbb{R}^{n+2}$ on a small diameter set is zero. We prove an additional generalization of the Borsuk-Ulam theorem for odd maps $S^{2n-1} to mathbb{R}^{2kn+2n-1}$. We prove such results for odd maps from the circle to any Euclidean space with optimal quantitative bounds. This in turn implies that any raked homogeneous trigonometric polynomial has a zero on a subset of the circle of a specific diameter; these results are optimal.
In the present paper we prove, that if the geodesic flow of a metric G on the torus T is quadratically integrable, then the torus T isometrically covers a torus with a Liouville metric on it, and describe the set of quadratically integrable geodesic flows on the Klein bottle.
For non-homotopic maps $u,vin C^{infty}(M,N)$ between closed Riemannian manifolds, we consider the smallest energy level $gamma_p(u,v)$ for which there exist paths $u_tin W^{1,p}(M,N)$ connecting $u_0=u$ to $u_1=v$ with $|du_t|_{L^p}^pleq gamma_p(u,v)$. When $u$ and $v$ are $(k-2)$-homotopic, work of Hang and Lin shows that $gamma_p(u,v)<infty$ for $pin [1,k)$, and using their construction, one can obtain an estimate of the form $gamma_p(u,v)leq frac{C(u,v)}{k-p}$. When $M$ and $N$ are oriented, and $u$ and $v$ induce different maps on real cohomology in degree $k-1$, we show that the growth $gamma_p(u,v)sim frac{1}{k-p}$ as $pto k$ is sharp, and obtain a lower bound for the coefficient $liminf_{pto k}(k-p)gamma_p(u,v)$ in terms of the min-max masses of certain non-contractible loops in the space of codimension-$k$ integral cycles in $M$. In the process, we establish lower bounds for a related smaller quantity $gamma_p^*(u,v)leqgamma_p(u,v)$, for which there exist critical points $u_pin W^{1,p}(M,N)$ of the $p$-energy functional satisfying $gamma_p^*(u,v)leq |du_p|_{L^p}^pleq gamma_p(u,v).$
The main result of this note is a parametrized version of the Borsuk-Ulam theorem. We show that for a continuous family of Borsuk-Ulam situations, parameterized by points of a compact manifold W, its solution set also depends continuously on the parameter space W. Continuity here means that the solution set supports a homology class which maps onto the fundamental class of W. When W is a subset of Euclidean space, we also show how to construct such a continuous family starting from a family depending in the same way continuously on the points of the boundary of W. This solves a problem related to a conjecture which is relevant for the construction of equilibrium strategies in repeated two-player games with incomplete information. A new method (of independent interest) used in this context is a canonical symmetric squaring construction in Cech homology with coefficients in Z/2Z.
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