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About 2 decades ago I heared from some one that there are infinitely many open contractible subsets of space mutually non homeomorphic to each other. I confess that I do not remember the deta …

The Shub conjecture on topological entropy $h(f)$ of self map f on manifold M says that the topological entropy is greater (or equal) than (to) the log of maximum absolute values of the eigenv …

What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis? Please see this related post and also the following post.. For Mathematical development …

I learned from the following post that the total space of the tautological line bundle over $\mathbb{R}P^{n}$ is diffeomorphic to $\mathbb{R}P^{n+1}\setminus \{pt\}$.(There is a natural embedding$([x …

In this question we consider the standard inclusion of $\mathbb{C}P^{n}\subset \mathbb{C}P^{n+1}$. It is well known that $\mathbb{C}P^n$ is not a retract of $\mathbb{C}P^{n+1}$. What about if we …

A topological space $X$ satisfies the "Periodic orbit property", briefly POP, if for every continuous map $f:X \to X$, there exist a natural number $n$ and a point $x_{0}\in X$ such that $f^{n}(x_{ …

The rank of an smooth manifold M is defined by Milnor, as follows: "The maximum number of independent commuting vector fields on M" For example it is well known that the rank of $S^{3}$ is 1 (Lima, …

Assume that we have a codimension one foliation of a manifold $M$ which is generated by a one form $\alpha$. So the following $\phi$ satisfies $\phi \circ \phi =0$:$$\phi:\Omega^{i}(M)\to \O …

Let we have a complex of abelian topological or lie groups $$\ldots \to G_{n}\to G_{n+1}\to \ldots$$ such that the image of $G_{n}$ is a closed subgroup of $G_{n+1}$. Then we have a complex of fu …

The definition of cohomology of a complex is based on the following: We have a complex (of appropriate objects) $$0\leftarrow C_0\leftarrow C_1\leftarrow C_2\ldots \leftarrow C_n\ldots$$ Then for an a …

For a self map $f$ on a topological space $X$ we replace "trace" with "determinant" in the alternative Lefschetz formula $$\Lambda(f)=\sum(-1)^i trace(f^*)|H^i(X,\mathbb{Q})$$ So we have $$\Lam …

What is an example of a nontrivial principal bundle whose fibre space $G$, total space $P$ and base space $M$ are compact connected manifolds (the fiber $G$ is a compact Lie group) such that $$H^*(P,\ …

Edit: According to comment of Prof. GoodWillie we revise the question. Put $M=GL(n,\mathbb{R})$. We identify $M_n(\mathbb{R})$ with $\mathbb{R}^{n^2}$: The identification is based on the lexicograph …

In the usual singular homology of a topological space $X$, one consider the free abelian group generated by all continuous maps from the standard simplex $\Delta^{n}$ to $X$. Now we can r …

Motivated by this paper and its economics motivations, we recall that a social choice among $n$ objects is a continuous function $$f:\overbrace{M\times M\times\cdots\times M}^{\text{$n$ times}}\to …