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 http://ifts.zju.edu.cn/profiles/xingangwang/Course2010/download/Yorke-chaos.pdf page 208, […]

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 […]

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 […]

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 […]

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: […]

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 […]

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 […]

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.

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 […]

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 […]

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