Articles of homotopy theory

Winding Numbers and Fixed Point Theorems

It is known that winding numbers can be used to prove the existence of fixed point theorems in two dimensions. We look at the vector $f(x)-x$ and if the corresponding winding number does not equal $0$ then we can establish the existence of a fixed point. For example, in the following book page 208, […]

What is a homotopy equivalence?

I have found the following problem. Let $A\subseteq X$ be a contractible space. Let $a_0\in A$. Is the embedding $X\setminus A\to X\setminus\{a_0\}$ a homotopy equivalence? I don’t understand the question. I know that two spaces $Y$ and $Z$ are homotopy equivalent when there are such continuous $f:Y\to Z$ and $g:Z\to Y$ that $f\circ g$ is […]

Computation of the hom-set of a comodule over a coalgebra: $Ext_{E(x)}(k, E(x)) = P(y)$.

First of all, since every other book somehow mentions that this is trivial, I apologize if it turns out that I am just misunderstanding something in the definitions. So here goes: The motivation for this is the proof that $Ext_{\Gamma}(k, k) = P(y_1, y_2, …)$ where $\Gamma$ is a commutative, graded connected Hopf algebra of […]

invariance of integrals for homotopy equivalent spaces

I just wanted to know whether the integral of a closed n-form is invariant if we integrate it over homotopy equivalent spaces. This seems like a generalization of “Homotopy invariance of line integral on manifolds”. Am I right? Something like: $$ \int_{\Sigma} \omega = \int_{\Sigma’} \omega$$ for homotopy equivalent $\Sigma\equiv \Sigma’$ Edit: Ok, I found […]

Homotopy equivalence induces bijection between path components

If $x\in X$, let $C(x)$ the path component of $x$ (the biggest path connected set containing $x$), and similarly if $y\in Y$. Let $C(X)$ and $C(Y)$ the family of all path components of $X$ and $Y$. Let $f:X\to Y$ be a homotopy equivalence. We define $G:C(X)\to C(Y)$ by $G(C(x))=C(f(x))$. Then I want to prove that: […]

Torus cannot be embedded in $\mathbb R^2$

I’ve shown that $T^2$ can be embedded in $\mathbb R^3$. I just can’t see why it can not be embedded in $\mathbb R^2$. Ideas: suppose $F: \mathbb S^1\times \mathbb S^1 \to \mathbb R^2$ is continuous injective then we can construct (somehow) $G:\mathbb S^1\to \mathbb R$ continuous injective so we get a contradiction. we know that […]

Every absolute retract (AR) is contractible

This is homework. I need to show that every AR is contractible. All I can basically do here is list definitions: A space $Y$ is AR if: $X$ is metrizable, $A$ is closed subset of $X$ and $f: A \mapsto Y$ is continuous, then $f$ has a continuous extension $g: X \mapsto Y$. A space […]

Can $S^0 \to S^n \to S^n$ become a fiber bundle when $n>1$?

As we know, $S^0\to S^n \to \mathbb RP^n$ is a fiber bundle with the usual covering space projection. I wonder if we can construct a projection $S^n \to S^n$ such that $S^0 \to S^n \to S^n$ becomes a fiber bundle. I need this result to complete the proof in here.

Understanding the inclusion of sets in the open category of X $Op_X$ and what \{pt\} denotes

What I am trying to understand is what is going on with the inclusion of sets, as if I understand correctly they are the morphisms of the category of open sets on X: $Op_X$ is the category of open sets such that for open $U,V \in X$: $\ Hom_{Op_X}(U,V) = \begin{cases} \{pt\} & U \subset […]

Contradictory; Homotopy equivalence and deformation retract problem

The question Show that $S^1$ is a deformation retract og $D^2\setminus\{(0,0)\}$ the unit punctured disc. The solution the inclusion map $i:S^2 \to D^2\setminus\{(0,0)\}$ and $$j:D^2\setminus\{(0,0)\} \to S^1: (x,y) \to \frac{a}{\sqrt{x^2+y^2}}(x,y)$$ are inverse homotopy equivalences via the straight line homotopy. Question marks on this one. I recall the definition of homotopy equivalences $X,Y$ are homotopy equivalent […]