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This is really just a comment/question for Tom (or for any other knowledgable topologist), but it has got far too long. It's also an attempt made from a position of almost complete ignorance to (re)co …

The compactified long closed ray $\overline R$ will have two endpoints, but these are distinguishable. One has a neighbourhood homeomorphic to $[0,1)$ and the other doesn't. This scuppers "long homoto …

I don't follow your comments about the projective plane. Surely the geometric realization of the simplicial complex consisting of $$\{a,b,c\},\{a,c,d\},\{a,d,e\},\{a,e,f\},\{a,f,b\},$$ $$\{b,c,e\},\{c …

The subgroup $j_* H_2(Q)$ must be generated by twice the generator of $H_2(P^2(\mathbb{C}))$ (I'm dropping the coefficient group from my notation). To see this, your map $\psi$ decomposes as the embed …

The group operation corresponds to the multiplication map $\mu:A\otimes A\to A$ and the identity should be the natural map $\iota:k\to A$. Both these should be coalgebra maps. The inverse should corre …

Does the following work? Let $A$ and $B$ be components of $\Omega X$ and assume that $A$ is the component containing the trivial path based at $x_0$. Let $f$ be any element of $B$. Then $f$ is a path …

The key word in this context is functor. The point is that homology, homotopy etc. are functors. For example consider homology $H_n$. This is a functor from the category of topological spaces to the c …

I don't quite see how to complete this argument, but here's an idea. Say your loop goes from 1 to $-1$ "along the top" and from $-1$ to 1 "along the bottom". A putative homotopy to the trivial loop is …

A lot of standard examples have $\pi_1(X)=C_2$ or have a $C_2$ as a factor. I think it's harder to visualize spaces with larger cyclic $\pi_1$, so here's a simple example. Pick a positive integer $n …

It's pretty standard to use $\bigwedge(V)$ or $\Lambda(V)$ for the exterior algebra on a vector space $V$ and $\bigwedge^k(V)$ or $\Lambda^k(V)$ for the $k$-th graded part. For symmetric algebras $S(V …

I'm afraid I know nothing about $B_3$ but here is my favourite construction of $\widetilde{\mathrm{SL}}_2(\mathbb{R})$. The elements of $\widetilde{\mathrm{SL}}_2(\mathbb{R})$ are pairs $(A,f)$ where …

An online search yielded a reference to Francis Sergeraert's paper, Triangulations of complex projective spaces, available at http://www-fourier.ujf-grenoble.fr/~sergerar/Papers/ . But, to quote the a …