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Let me make a few comments relating the various classical constructions you mentioned. The $BGL(R)^+$ construction produces an infinite loop space. It's constructed by taking $BGL(R)=\varinjlim BGL_n(R)$ and then attaching $2$-cells to kill the commutator subgroup of $\pi_1$ and attaching $3$-cells to preserve the original homology. It might seem ad-hoc ...


2

$U$ is open and contains $0$, hence contains some open interval $(-a,a)$ with $a>0$. Pick $n>\frac1a$. Then $I:=[\frac1{n+1},\frac1n]\subset (-a,a)\subseteq U$. From $g(\frac1{n+1})=\frac1{n+1}$ and $g(\frac1n)=\frac1n$, it follows by the Intermediate Value Theorem (viewing $g$ as a function $I\to\Bbb R$) that $g(\xi)=\frac{\frac1{n+1}+\frac1n}{2}$ for ...


1

You have to apply Munkres' Theorem 22.2. Consider the map $g = p \circ h : S^1 \to S^1$. Each set $p^{-1}(z)$ consists of two antipodal points $w_1, w_2 = -w_1$ (these are the two complex square roots of $z$). But obviously $g$ is constant on all $p^{-1}(z)$ since $g(w_2) = h(w_2)^2 = h(-w_1)^2 = (-h(w_1))^2 = h(w_1)^2 = g(w_1)$.


1

The setup is that there is an embedding $j:X\hookrightarrow\mathbb{R}^n$, an open set $Y\subset\mathbb{R}^n$ with $X\cong j(X)\subseteq Y$, and a continuous map $r:Y\rightarrow X$. The question regards the topology on $Y$. Suppose that $V\subset Y$ is any subset which is open in $Y$. Because $Y$ has the subspace topology inherited from $\mathbb{R}^n$, this ...


1

I learned this fact from A. Dold's book Lectures on Algebraic Topology. If you are interested, you will find there a much more general statement. With the ENR $X$ given, fix an embedding $j:X\hookrightarrow\mathbb{R}^n$, and a retraction $r:N\rightarrow X$, from an open neighbourhood $N$ of $X$ in $\mathbb{R}^n$. Now let $U\subseteq X$ be open and $x_0\in U$....


1

I think you are right in following intuitions at the zero level, but I strongly recommend you not to use simplex by simplex constructions and use abstract stuff to create maps, as you were doing. My first intuition was the following (see HTT 1.2.2 for definitions): $\mathcal{E}$ can be contracted to the cone point in the first component, yielding $\text{Hom}...


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