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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.
Jun
2
comment Universal property of universal bundles.
Okay, fair enough. I should have read more carefully...
Jun
1
answered Universal property of universal bundles.
Jun
1
comment Universal property of universal bundles.
Dylan, this argument seems circular. How do you know that the map classifying this bundle on XxI has anything to do with the maps $f$ and $g$, unless you've already proven that there's only one map, up to homotopy, classifying the bundles $f^* EG$ and $g^* EG$?
Jun
1
comment Papers in algebraic topology
books.google.com/books/about/…
May
16
comment Functoriality of the Fundamental group
Good places to look for this material include Segal's paper Classifying spaces and spectral sequences and Milnor's paper Geometric realizations of semi-simplicial sets.
May
16
comment Why base point makes a huge difference?
Yeah, Dover is really great.
May
16
comment Lemma of Whitehead
The space $X\cup_{\phi_i} e^\lambda$ is the quotient space of the disjoint union of $X$ and $e^\lambda$ formed using the equivalence relation $x\sim e$ if $e$ is in the boundary of the cell $e^\lambda$ and $\phi_i (e) = x$. So to check that $k$ is well-defined you just need to show that it respects these identifications. Incidentally, this is a page from Milnor's Morse Theory book, I'm pretty sure.
May
16
comment Spectral sequences: equivalence of exact couples and classic (?) method
Possibly what you're looking for is also written out in detail in Lectures 6-7 here: math.nmsu.edu/~ramras/643.html Probably there's not much difference between what I wrote and what's in McCleary's book. I don't think I had his book with me when I was writing those notes, so I guess I never checked.
May
16
comment Why base point makes a huge difference?
Mosher and Tangora was recently reprinted!
May
16
comment Functoriality of the Fundamental group
To put what Zhen Lin wrote in more explicit form, given a group homomorphism $f: G\to H$, there is an induced map $Bf: BG\to BH$. It is elementary to check that $(Bf)_*: \pi_1 BG\to \pi_1 BH$ agrees with $f$ under the canonical isomorphisms $G\simeq \pi_1 BG$ and $H\simeq \pi_1 BH$. It seems to me that this is all the question asked for.
May
16
comment Functoriality of the Fundamental group
Drew, I think the question was about discrete groups, not topological ones. Also, usually E-M space just means a space with exactly at most non-zero homotopy group, so K(Z,2) is an E-M space.