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In standard topological terms, the exact sequence that relates homotopy groups of the base $B$, fiber $F$ and total space $E$ of topological fibration gives $$\pi_1(F) \to \pi_1(E) \to \pi_1(B) \to \ …

You can do a reverse construction: start with a sphere without 4 points; now add two points over each one in such a way that every time you go around one hole the two points get interchanged. The sam …

Origin This question was asked by John Baez in This Week's Finds in Mathematical Physics (Week 286). Therefore, please don't upvote this question (unless you really want to), but do upvote the answer …

I'm rereading my notes and they mention that $K_{23}(\mathbb Z) = \mathbb Z/(65520)$ This looks like a good point to stop and ask whether there is any explanation for this $K$-group of integers (23 i …

There is a followup article, first in the 5-article series Derived Algebraic Geometry: Stable Infinity Categories, arXiv:math/0608228 In particular, The theory of stable ∞-categories is not re …

From a post to The Jouanolou trick: Are all topologically trivial (contractible) complex algebraic varieties necessarily affine? Are there examples of those not birationally equivalent to an affin …

I might be missing something about question 3. Here's a simple construction: Consider a projective space $P$ of dimension $\text{dim}\\, P_1 + \text{dim}\\, P_2$ that contains both $P_1$ and $P_2$ in …

For a fixed $d$, is there a relationship between the homotopy groups of smooth $d$-manifolds? The $d=1$ case is trivial, but I already don't know how to approach $d=3$ (I should have said that th …

To study the topology of Lie groups, you can decompose them into the simple compact ones, plus some additional steps, such as taking the cover if necessary. After that, the structure of $SO(n)$'s is r …

I don't know for sure, but it would appear he means that it's easier to construct a cubic chain on a product $X \times Y$ given cubic chains on $X$ and $Y$ compared to the simplex chain given two simp …

This came up in the question about Eilenberg-MacLane spaces. Given the definition of K(G, n), it's easy to prove that there is a map K(G,n) x K(G,n) --> K(G,n) that endows cohomology with an additive …

Indeed, the statement is that homotopy classes of continuous maps of pointed spaces $[X, K(G, n)]$ are in 1-1 correspondence with the elements of singular homology $H^n(X, G)$ for a CW-complex $X$. T …

One example would be a map induced by a morphism $f: X \to Y$ in the long homology sequence. E.g. suppose the top row is a cohomology of pair $(X, A)$ and the bottom row is the cohomology of pair $( …

I think the proper Whitehead for maps says that if the cone of the map has trivial homotopy groups, then the map is a homotopy equivalence. Edit: see also the discussion of Whitehead theorem in the c …

I don't remember exactly, but I think your question and in general your approach to topology should be along the lines of Bott-Tu book.