Intereting Posts

When is the closed unit ball $B^*$ in the dual space strictly convex?
Do we really need reals?
Sphere-sphere intersection is not a surface
Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$
Soviet Russian mathematics books
Visual explanation of the indices of the Riemann curvature tensor
What is the integral of 0?
Prove that if $R$ is von Neumann regular and $P$ a prime ideal, then $P$ is maximal
Dominos ($2 \times 1$ on $2 \times n$ and on $3 \times 2n$)
How to put 9 pigs into 4 pens so that there are an odd number of pigs in each pen?
Hamiltonian for Geodesic Flow
How can I (algorithmically) count the number of ways n m-sided dice can add up to a given number?
Proof that convex open sets in $\mathbb{R}^n$ are homeomorphic?
An elementary functional equation.
prove that $\frac{(2n)!}{(n!)^2}$ is even if $n$ is a positive integer

We counteract this answer by adding the *rigidity assumption*: Is there still a counterexample?

Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that:

$X_{n}$ is a regular$^1$ CW-complex of constant local dimension$^3$ $n$, it is of finite type$^4$, boundaryless$^2$, unbounded, uniform$^5$, and it is the $n$-skeleton of $X_{n+1}$, which is n-connected. Moreover, the distances $d_{n}$ , $d_{n+1}$ generate the same topology on $X_{n}$ and $\forall x,y \in X_{n} \ d_{n+1}(x,y) \le d_{n}(x,y)$.

Finally $(X_{n},d_{n})$ is quasi-isometric to $(X_{n+1},d_{n+1})$, through the inclusion map $X_{n} \subset X_{n+1}$, and a distance $d$ on $ \bigcup{X_{n}}$ is defined (for $x, y \in X_{n_0}$) by $d(x,y) := lim_{n (\ge n_0) \to \infty} d_{n}(x,y)$.

**Rigidity assumption**: if $S$ is a connected subspace of $X_{n}$ such that $S$ contains the geodesic paths between all its points, for $d_{n}$ **and** $d_{n+1}$, then $d_{n} = d_{n+1}$ on $S$.

- Homotopy equivalence iff both spaces are deformation retracts
- Local coefficients involved in the obstruction class for a lift of a map
- Top homology of a manifold with boundary
- Homology of surface of genus $g$
- Fundamental Polygon of Real Projective Plane
- Fundamental group of mapping torus?

*Definition* : Let $X:=\overline{\bigcup{X_{n}}}$ be the completion of the metric space $\bigcup{X_{n}}$ with $d$.

**Question** : Is $X$ weakly contractible ?

*Remark:* Some of these conditions could be useless for a proof, and others, highly generalized.

*Motivation*: See here for applications to *geometric group theory* and *noncommutative geometry*.

$^1$Regular (for a CW complex) : the attaching maps are homeomorphism (see this post).

$^2$Boundaryless (for a regular CW complex) : the boundary of each closed cell is contained is the union of the boundaries of other closed cells.

$^3$Constant local dimension : the topological dimension of all neighborhood of all point, is constant.

$^4$Finite type : finitely many $r$-cells ending in a fixed $(r-1)$-cell.

$^5$Uniform : For all $r$-cell $c_{1}$ and $c_{2}$, there is a neighborhood $n_{1}$ of $c_{1}$ and $n_{2}$ of $c_{1}$, such that $n_{1}$ is homeomorphic to $n_{2}$.

- How many sets can we get by taking interiors and closures?
- Proving that the join of a path-connected space with an arbitrary space is simply-connected
- Proof that every metric space is homeomorphic to a bounded metric space
- Accumulation points of the set $S=\{(\frac {1} {n}, \frac {1} {m}) \space m, n \in \mathbb N\}$
- Can contractible subspace be ignored/collapsed when computing $\pi_n$ or $H_n$?
- Finding a space $X$ such that $\dim C(X)=n$.
- Metric space is totally bounded iff every sequence has Cauchy subsequence
- What (and how many) pieces does the Banach-Tarski Paradox break a sphere into?
- Is Alexandroff Duplicate normal?
- Yoneda's lemma and $K$-theory.

- Prove any orthogonal 2-by-2 can be written in one of these forms…
- How to prove a trigonometric identity
- How to compute the following Jacobian
- Weak Law of Large Numbers for Dependent Random Variables with Bounded Covariance
- Does every infinite field contain a countably infinite subfield?
- The case of Captain America's shield: a variation of Alhazen's Billard problem
- Why principal ideals in Z are not maximal?
- Do the Laurent polynomials over $\mathbb{Z}$ form a principal ideal domain?
- Why are these two definitions of a perfectly normal space equivalent?
- Prove that $a^2+ab+b^2\ge 0$
- Computing $\int^1_0 \frac{p^2-1}{\text{ln } p}dp$
- Mathematical difference between white and black notes in a piano
- Is this determinant always non-negative?
- Complex finite dimensional irreducible representation of abelian group
- On the closedness of $L^2$ under convolution