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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 …

Assume that $X$ is a path connected hausdorff topological space. Let $\alpha\in C^{1}(X,\mathbb{R})$ and $\beta\in C^{n}(X,\mathbb{R})$ be cochains in real singular cohomology. Asume that $\alpha \s …

Let $M$ be the manifold of all matrices in $M_{n}(\mathbb{R})$ with fixed rank $0<k<n$. The projectivization of $M$ is denoted by $PM$. Does $PM$ satisfy fixed point property?

Let $f$ be a continuous function on complex Grassmannian $G(k, 2n+1)$. Is it true to say that there is a $k$-plane $Y$ such that $Y$ has nontrivial intersection with $f(Y)$? A motivation for …

Let $X$ be a compact connected Hausdorff topological space. We say $X$ is a cohomologicaly minimal space, briefly CM space, if $X$ satisfies the following property: "For every proper subset $A\ …

Let $G$ and $H$ be two topological groups. Assume that $\phi:\pi_{1}(G) \to \pi_{1}(H)$ is a group homomorphism. Is there a continuous function $f:G\to H$ such that $f_{*}=\phi$?

Assume that $n>1$. The configuration space of $S^n$ is defined as follows $$M_n=\{(x,y)\in S^n\times S^n\mid x \neq y\}$$ We have two questions: 1.Is there a continuous function $f:M_n \ …

Is there a continuous map $f:\mathbb{C}P^n \to \mathbb{C}P^m$, for some $n>m$ which preserve orthogonality?Namely $x\perp y \implies f(x) \perp f(y) $? If yes, are there two non homotopic …

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 …

I asked this question in MSE, but I did not received any answer, so I repeat it here: https://math.stackexchange.com/questions/858238/a-question-on-fixed-point-property Assume that $0<k<n-1$, Note …

Let $G$ be a compact Lie group with invariant measure $\mu$. An odd function is a continuous function, $\phi:G\to \mathbb{C}$, such that $\int_{G} \phi d\mu=0$. An odd map is a continuous map, $f:G\to …

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 …

For a topological space X we can consider the coefficient of singular cohomology in a Lie algebra A. Then we obtain a graded Lie algebra, that is [x,y]=(-1)^i+j-1 [y,x], for homogeneous elemen …

Acording to the comment of Mark Grant and the answer of Ryan Budney, I revise the question: For what even $n$, there is a retract embedding of of $S^n$ in its unit tangent bundle?

The following paper and its references contains some algebraic consequences of vector bundle theory. Vakhtang Lomadze, Applications of vector bundles to factorization of rational matrices, Line …