I have re-written the content at *differentiable manifold*, trying to make it look a little nicer. Also gave *topological manifold* some minimum of content.

I am working on the entry *topological manifold*.

I gave it a subsection *locally Euclidean spaces*, which maybe eventually wants to be split off as an entry in its own right.

Now I have added statement and proof that locally Euclidean spaces are $T_1$, sober and locally compact (in the compact neighbourhood base sense): here.

]]>the first paragraphs at *topological vector space* seem odd to me.

I’d think it should start out saying that a topological vector space is a vector space *over a topological field* $k$, such that etc.. Then the following remark presently in the entry, about the internalization using the discrete topology is moot.

Created a little entry *Vect(X)* (to go along with *Vect*) and used the occasion to give *distributive monoidal category* the Examples-section that it was missing and similarly touched the Examples-section at *rig category*.

There is a question about the education curriculum (the most advanced study) on algebraic topology

I am interested in what topics and to which year of study a person should know who wants to make a big contribution to AT, higher category theory

Example:

2-3 course - Spectrum, algebraic K theory, model categories, stable homotopy theory, homology theory, Stolz-Teichner programme

Thank you ]]>

I added the definition and several references on higher dimensional knots under knot.

]]>added a little bit to *foliation*: a brief list of equivalent alternative definitions and and Idea-section with some general remarks.

I would like to know the great nForum community of "categorical physics" would be interested in contribute. Give it a chance and take a look into the project https://github.com/gcarmonamateo/GeomFormes and hopefully caught your interest in it.

(Sorry for the imprecisions in the English language). ]]>

gave the old entry *wedge sum* its explicit formal definition. Also added two examples.

tried to polish one-point compactification. I think in the process I actually corrected it, too. Please somebody have a close look.

]]>A student asked “What is a cobordism?” and I checked and realized that the $n$Lab entry *cobordism* was effectively empty.

So I have now added some basic text in the Idea-section and added a bare minimum of references. Much more should be done of course, but at least now there are pointers.

]]>I have given *pseudogroup* an entry of its own, for the moment just copying there the definition from *manifold*. This is so as to be able to add references for the concepts, which I did.

I have spelled out the proofs that over a paracompact Hausdorff space every vector sub-bundle is a direct summand, and that over a compact Hausdorff space every topological vector bundle is a direct summand of a trivial bundle, here

]]>I have created an entry on the *quaternionic Hopf fibration* and then I have tried to spell out the argument, suggested to me by Charles Rezk on MO, that in $G$-equivariant stable homotopy theory it represents a non-torsion element in

for $G$ a finite and non-cyclic subgroup of $SO(3)$, and $SO(3)$ acting on the quaternionic Hopf fibration via automorphisms of the quaternions.

I have tried to make a rigorous and self-contained argument here by appeal to Greenlees-May decomposition and to tom Dieck splitting. But check.

]]>Expansion of references section at differential topology.

]]>wrote out statement and proof that *locally compact and sigma-compact spaces are paracompact*

I gave *locally compact topological space* an Idea-section and added the other equivalent definition (here).

I have been making trivial edits (adding references, basic statements, cross-links ) to *Hopf invariant* and a bunch of related entries, such as *Kervaire invariant*, *Hopf invariant one problem*, *Arf-Kervaire invariant problem*, *normed division algebra*.

Is the Strøm model category left proper? I know that pushout along cofibrations of homotopy equivalences of the form $A \to \ast$ are again homotopy equivalences. (e.g. Hatcher 0.17) Maybe the proof directly generalizes, haven’t checked.

]]>I have edited a bit at *general topology*, trying to stream-line for readability.

brief entry *extremally disconnected topological space*

At *Fréchet space* I have added to the Idea-section a paragraph motivating the definition via families of seminorms from the example of $\mathbb{R}^\infty = \underset{\longleftarrow}{\lim}_n \mathbb{R}^n$. And I touched the description of this example in the main text, now here.

added in CW-complex in the Examples section something about noncompact smooth manifolds.

Eventually it would be good to state here precisely Milnor’s theorem etc. Googling around I seem to see a lot of misleading imprecision in the usual statements along these lines (on Wikipedia and MO) concerning the distinctions between countably generated and general CW-complexes and concerning homotopy equivalence vs weak homotopy equivalence.

]]>For better readability, I have split off *proper map* (topology) from *proper morphism* (general) and added disambiguation. Added classes of examples at *proper map*.

at *Serre fibration* I have spelled out the proof that that with $F_x \hookrightarrow X \overset{fib}{\to} Y$ then $\pi_\bullet(F) \to \pi_\bullet(X)\to \pi_{\bullet(Y)}$ is exact in the middle. here.

(This is intentionally the low-technology proof using nothing but the definition. )

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