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Homotopy theory, homological algebra, algebraic treatments of manifolds.

46 votes
Accepted

If $X$ and $Y$ are homotopy equivalent, then are $X \times \mathbb{R}^{\infty}$ and $Y \time...

This question is answered by two classical theorems of infinite-dimensional topology, which can be found in the books of Bessaga and Pelczynski, Chigogidze or Sakai. Factor Theorem. For any Polish abs …
Taras Banakh's user avatar
  • 41.8k
19 votes
Accepted

A parametric version of the Borsuk Ulam theorem

Theorem. For a topological space $X$ the following conditions are equivalent: 1) for any continuous map $f:X\times S^2\to\mathbb R^2$ there exists a point $s\in S^2$ such that $f(x,s)=f(x,-s)$ …
Taras Banakh's user avatar
  • 41.8k
19 votes
Accepted

Does $\mathbb C\mathbb P^\infty$ have a group structure?

I noticed that this question still has no accepted answer and all existing answers are rather long. It seems that the answer can be easily obtained using some results of infinite-dimensional topology, …
Taras Banakh's user avatar
  • 41.8k
17 votes

Contractible topological groups

If a topological group is contractible, then it is locally contractible (using the group operation produce a contraction which does not move the unit of the group). By a classical result of [A. Gleaso …
Taras Banakh's user avatar
  • 41.8k
14 votes

What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simpl...

The simply connected deletion number equals the continuum. This follows from the fact that for any dense subset $A$ in the real line the subset $A_3=(A\times \mathbb R\times \mathbb R)\cup (\mathbb R\ …
Taras Banakh's user avatar
  • 41.8k
11 votes
Accepted

Space with compactly closed diagonal but which is not weak Hausdorff

The space $X$ constructed in Theorem 1.5 of this preprint has the required properties. This space contains a non-closed compact metrizable subspace $K$, so is not weakly Hausdorff. On the other hand, …
Taras Banakh's user avatar
  • 41.8k
8 votes
Accepted

Do solenoids embed into Möbius strips?

No solenoid can be embedded into the Mobius strip. To derive a contradiction, assume that some solenoid $S$ embeds into the Mobius strip $M$. Let $\pi:C\to M$ be a 2-fold covering map of the cylinder …
Taras Banakh's user avatar
  • 41.8k
8 votes
2 answers
2k views

Classification of closed 3-manifolds with finite first homology group?

I am interested in a topological classification of connected closed 3-manifold $M$ that have finite homology group $H_1(M)$. Since $H_1(M)$ is the abelization of the fundamental group $\pi_1(M)$, ea …
Taras Banakh's user avatar
  • 41.8k
7 votes
0 answers
119 views

The automorphism group of the fibered cylinder

My collegue (Oleg Gutik) is interested in finding a proper reference to a description of the group $G$ of homeomorphisms $h:\mathbb T\times\mathbb R\to\mathbb T\times\mathbb R$ of the cylinder that pr …
Taras Banakh's user avatar
  • 41.8k
6 votes
Accepted

Are infinite simplicial complexes all manifolds?

This question concerns manifolds modeled on the direct limit $\mathbb R^\infty$ of Euclidean spaces. The theory of such manifolds is well-developed. The most important results of this theory were obta …
Taras Banakh's user avatar
  • 41.8k
5 votes
Accepted

Must an inverse limit of simply connected groups be simply connected?

It seems that the inverse limit of simply connected Lie groups is simply connected. The argument uses the well-known fact that the second homotopy group of any Lie group is trivial (see e.g. Homotopy …
Taras Banakh's user avatar
  • 41.8k
5 votes
Accepted

Lifting local compactness to a covering space

A suitable counterexample can be constructed as follows. Let $\mathbb I=\{x\in\mathbb R:0<x<1\}$ denote the open unit interval and let $B=\{0\}\cup(\omega\times\mathbb I)\cup\{1\}$ be endowed with th …
Taras Banakh's user avatar
  • 41.8k
5 votes
Accepted

A question on Möbius strip and Jordan curve

It seems that the answer to this problem is affirmative. We can argue as follows. Assume that $A\subset\mathbb R^2$ is a subspace whose symmetric square $A^2/_\sim$ is homeomorphic to the Mobius stri …
Taras Banakh's user avatar
  • 41.8k
5 votes

Retracting off a compact set

Let us show how to find such a retraction for $n=2$ (I do not know if this method generalizes to higher dimensions). Given a compact set $C\subset\mathbb R^2$ and an open neighborhood $U\subseteq\mat …
Taras Banakh's user avatar
  • 41.8k
4 votes
Accepted

Inscribing a "chain" into an open cover

The answer is ``yes'' if $X$ is Hausdorff. We can identify the arc $C$ with the unit interval $[0,1]$ and assume that $[0,1]$ is contained in $X$ and is covered by a family $\mathcal U$ of open subs …
Taras Banakh's user avatar
  • 41.8k

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