Tagged Questions

605 views

When are maps between topological spaces homotopic?

I wanted to ask if there is any known mehod to quantify 'how many' homotopy classes of maps there are between two given topological spaces $X$, $Y$ (CW-complexes, say). So far I had the following ...
361 views

Are loop spaces of homotopically equivalent spaces homotopically equivalent? [closed]

Let $f:X \to Y$ be a homotopy equivalence of pointed topological spaces. Then, is the induced map of pointed loop spaces $\Omega (f): \Omega X \to \Omega Y$ a homotopy equivalence? Here, loop spaces ...
228 views

Homotopy Equivalences and Induced Correspondences between Fibre Bundles

Suppose that $f:X\rightarrow Y$ is a homotopy equivalence of manifolds. Given a manifold $F$, the pullback construction for $f$ yields a correspondence between isomorphism classes of fibre bundles ...
179 views

Are constructible derived categories invariant up to weak homotopy equivalence?

Let $X$ and $Y$ be two topological spaces and $R$ be commutative ring. Let $D_c^b(X, R)$ and $D_c^b(Y,R)$ be their respective bounded derived categories of constructible sheaves of $R$-modules. I ...
182 views

( Homotopy) Y ENR and contractible subset => Y is a retract

I'm trying to solve the following question: Y $\subset R^n$ is a euclidian neighborhood retract. I want to prove that if $Y$ is contractible it is a rectract of $R^n.$
Given two complex line bundles over the complex projective line ${\mathbb CP}^1$ prove or disprove that their total spaces are homeomorphic if and only if their Chern numbers are equal up to sign. ...
Are there topological spaces with non trivial homotopy groups $\pi_{>=2}$ that are visualizeable? I'm thinking about the the shape of what is called a "higher dimensional hole" but I can't find ...