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awarded  Good Question
Apr
30
awarded  Popular Question
Apr
27
comment Is the de Rham complex a free (commutative?) differential graded algebra?
I think, but I haven't thought it out much, that the de Rham dga is the free skew-commutative graded $C^\infty(M)$-dga with $\Omega^1(M)$ in degree 1. Doesn't one deduce this from proposition 2.6 (page 57) in "Elements of noncommutative geometry" by Varilly et al.? One avoids problems of the kind pointed by Eric Wofsey by actually starting from the 1-forms.
Apr
5
awarded  Nice Question
Mar
22
comment Geometric realization of function complexes of simplicial sets
Schwede theorem A.3.2 asserts your suspicion, though he doesn't really give either references or proofs.
Mar
19
comment Kan's loop group construction
I like Stevenson's "Décalage and Kan's simplicial loop group functor" and some references therein. You can also look at May's "simplicial objects in algebraic topology". I also believe they're discussed in Curtis' "simplicial homotopy theory" survey paper.
Feb
19
comment Fat geometric realization weakly equivalent to the usual one
By the way, the cellular chain complex of $|Sing(X)|$ coincides with what is usually called the "normalized singular chain complex" of $X$. This is like the usual singular chain complex, except that you're quotienting out by the images of degeneracies. I.e., when passing from the free simplicial abelian group on the singular set to chain complexes, you use the normalized Moore functor instead of the alternating face map functor.
Feb
10
comment Morphism induced by a cellular map between CW-complexes
I found by chance this other question that has the "homological" approach to induced maps.
Feb
2
awarded  Popular Question
Feb
2
comment Are there spectral sequences for calculating homology or cohomology of homotopy (co)limits?
Nice answer. It should be noted that a generalized homology theory carries filtered homotopy colimits to colimits (see here), so the spectral sequence is of use when considering non-filtered ones. Moreover, when one takes a homotopy pushout, this spectral sequence collapses to an exact sequence which is exactly Mayer-Vietoris, see here.
Jan
31
comment Morphism induced by a cellular map between CW-complexes
So when, in the answer, I said "let $\varphi_\alpha$ be the attaching map of $\alpha$", I was being sloppy.
Jan
31
comment Morphism induced by a cellular map between CW-complexes
I should note that Bertram's comment is spot on. You need to be careful. Choose a set of attaching maps for all cells of $X$. This choice defines: an isomorphism as the one you claim, the boundary map on $C_*^{cell}$, and induced maps on $C_*^{cell}$. Using the same set of attaching maps for all three choices gives you a natural isomorphism of chain complexes, where $H_n(X^n,X^{n-1})$ gets the boundary map I alluded to in my first comment above. So you have to be careful, because a CW complex does not come, in the usual definitions, with the attaching maps as part of the data.
Jan
31
awarded  Revival
Jan
30
revised Morphism induced by a cellular map between CW-complexes
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Jan
30
revised Morphism induced by a cellular map between CW-complexes
added 124 characters in body
Jan
30
answered Morphism induced by a cellular map between CW-complexes
Jan
30
comment Morphism induced by a cellular map between CW-complexes
Ok, I looked at Switzer again and I think what you want is proposition 10.13, though the formula is a bit convoluted. I also empathize with @Bertram 's point and point to these notes for a nice explanation of the phenomenon.
Jan
30
comment Morphism induced by a cellular map between CW-complexes
I empathize with your question, but I doubt that there will a direct expression, i.e. one that does not pass by the identification of $C_n^{cell}$ with $H_n(X^n,X^{n-1})$, if only for the simple reason that your cellular boundary maps are secretly passing by that interpretation already: they are the maps that correspond to the connecting homomorphism of the triple $(X^n,X^{n-1}, X^{n-2})$ under the natural isomorphism you're quoting. By the way, I think there is a nice exposition of this in Switzer, starting from 10.6.
Jan
22
awarded  Nice Question
Jan
13
comment Reduced homology group of wedge sum
Be careful, I think that -exactly- the point is not to use additivity. The additivity axiom (for a reduced theory) says exactly what you're trying to prove. The point here is that additivity for finite wedge is included in the Eilenberg-Steenrod axioms (not so for infinite wedges).