Dan Ramras
Reputation
1,306
Next privilege 2,000 Rep.
 Feb 29 comment Good pair vs. cofibration I just noticed that my previous comment is more or less what May says in Section 14.2 of his book; he's working with an arbitrary generalized homology theory. Feb 29 comment Good pair vs. cofibration Practically speaking, the important thing is to know that a cofibration induces a LES on homology (involving the homologies of A, X, and X/A). This can be seen by noting that the the inclusion of CA into the mapping cone Ci is a cofibration (it's obtained from i by a pushout) and $Ci\rightarrow Ci/CA = X/A$ is a homotopy equivalence (Hatcher 0.17). If Mi is the mapping cylinder, then (Mi, A) is good and gives the LES. More specifically, one sees in this manner that the natural map $H_*(X,A) \rightarrow H_* (X/A, A/A)$ is an isomorphism for any cofibration. Feb 29 comment Good pair vs. cofibration @StephenDedalus I don't understand the comment. It seems like this is not really addressed in May or Hatcher, and I don't see anything helpful at the linked answer or in the blog post linked to therein. (That blog post says something about U deformation retracting to A, but just as in May's book that's not really part of the statement.) Arne Strom's papers Notes on Cofibrations I and II say more clearly that U is deformable to A in X rather than saying that U deformation retracts to A. Feb 3 comment Example of Spherical Element (Simplicial Homotopy) @yoyostein: that is essentially a rephrasing of the definition of "spherical". The definition states that each maximal face of x (that is, d_i x) is equal to the basepoint * (which, again, really means d_0 d_0 ... d_0(*) if * lies in X_0). But then if you take any face of * you just get * again, so any sequence of face maps applied to a spherical element results in the basepoint. Feb 3 answered Example of Spherical Element (Simplicial Homotopy) Aug 21 awarded Yearling Feb 12 comment Flat vector bundles and constant transition functions S.S., you might find the appendix to my article arXiv:0710.0681 helpful in regards to your second question (to which the answer is "Yes"). It contains a discussion of the standard facts about the holonomy correspondence between (gauge equivalence classes of) flat connections and (conjugacy classes of) representations of the fundamental group. Oct 14 comment Face post of a subcomplex complement Cross-listed on MO: mathoverflow.net/questions/184324/… Sep 24 awarded Autobiographer Sep 21 comment What is a principal orbit It's worth noting that the terminology here ("maximal") is referring to the size of the orbit, rather than the size of the stabilizer. This makes the second paragraph above sound a little awkward (everything has bigger type than the maximal orbit?). Bredon does indeed define the maximal orbit to be the one with the smallest stabilizer (hence the largest orbit). Aug 21 awarded Yearling Apr 15 comment Homology Whitehead theorem for non simply connected spaces @OlivierBégassat: You're right. This is the basepoint issue I was worried about (and I didn't think through it carefully). McDuff and Segal just claim that the map $[S^1, B\Sigma_\infty] \to [S^1, B\Sigma_\infty]$ induced by $\sigma$ is an isomorphism, but this is not the same as the map on fundamental groups, of course. Apr 15 comment Homology Whitehead theorem for non simply connected spaces I think that a counterexample is given at the top of p. 281 in McDuff and Segal's article on group completion (available here: maths.ed.ac.uk/~aar/papers/mcdsegal.pdf) but maybe there is an issue of basepoints. They consider the shift map on the infinite symmetric group (finitely supported permutations of $\mathbb{N}$). Feb 13 comment Domain invariance for smooth functions I don't recall ever seeing a proof apart from proofs of the full invariance of domain theorem for continuous functions. Where did you see this referred to as "easy"? I would love to see an easy proof! Jan 27 awarded Disciplined Sep 29 awarded Popular Question Sep 17 comment Russell's Paradox Shouldn't it be said that the barber shaves exactly those men who do not shave themselves? Otherwise, it's not paradoxical for the barber to shave himself and still be a man. Aug 21 awarded Yearling Jul 13 comment Loop space and $K$-theory Also, I don't understand the first question, really. You're asking how to prove Bott Periodicity without the Yoneda Lemma. What proofs of Bott Periodicity do you know? There are many, and they all involve significantly deeper mathematics than the Yoneda Lemma. Jul 13 comment Loop space and $K$-theory For the second question to make sense, you need to specify a cell structure on both $BU$ and on $\Omega^2 (BU)$ (note that the factor of $\mathbb{Z}$ doesn't affect the based loop space). I don't think you're likely to find canonical CW structures on these spaces. I'm not even sure they're homeomorphic to CW complexes. They are certainly homotopy equivalent to CW complexes, although this is not obvious, especially for the loop space. There are several ways to show that a colimit like $BU$ is homotopy equivalent to a CW complex; for the loop space, you need an old theorem of Milnor.