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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 …

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 …

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 …

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 …

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 …

Edit: Accoring to the comment of Asura Path I revise the question. Let $X$ be a vector field on a Riemannian manifold $(M,g)$. So we have a $1$-form $\beta$ with $\beta(Y)=\langle Y,X\rangle …

Let $G$ be a topological group and $X$ be a topological space. Let $\alpha$, $\beta:G\times X\to X$ be two group actions. We say that these two actions are homotopic actions if there is a continuous p …

In your question you mentioned the word "Fredholm index". So I would like to say that in the circle case there are two different interpretations of Fredholm index of certain lin …

Is there a topological space $X$, which is not a singleton, and satisfies the following property? For every continuous function $f: X\times S^2\to\mathbb{R}^2$ there exist a point $x\in S^2$ such th …

At the right column of the page 654 of the paper, R. Palais, The Visualization of Mathematics: Towards a mathematical Exploratorium, Notice AMS it is written "There can be no eversion of …

Assume that $V$ is a finite dimensional real or complex normed linear space. Let $Iso(V)\subset GL(V)\subset L(V)$ be the space of linear isometric endomorphisms, invertible endomorphism and li …

Is there a natural number $n$, a compact Lie group $G$ of dimension less than $n$ and a continuous map $f:S^n \to G$ with $f(-x)=f(x)^{-1}$, such that $f$ is not a null homotopic map? This quest …

The group cohomology of a group $G$ is defined as the derived functor associated to the following left exact functor: $$FIX: \mathcal{M_G} \to \mathcal{Ab}$$ where $FIX$ is the functor from the catego …