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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 …

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 …

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 \ …

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 …

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.

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 …

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 …

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 …

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 …

Kind of late to the party, but the (weak) contractibility follows from $\pi_i(S^\infty) = 0$ for $i>0$.

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 …

See, e.g., Baez: http://math.ucr.edu/home/baez/octonions/node8.html or even better http://math.ucr.edu/home/baez/week173.html

Filling the gaps in my knowledge to understand the tmf question. So, what is the analogue of elliptic curve over the category of spectra?

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 …

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 …