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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
deleted 751 characters in body
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).
Jan
6
awarded  Good Question
Jan
2
comment A module over an algebra. Is it a vector space?
Another thought: one could think that one could make a "relative" version of an $A$-module, as in that post: an $A$-module relative to $k$ should be a $k$-module $M$ with an action of $A$ such that the action of $k$ it induces coincides with the original one. But it's a boring notion, unlike the bimodule case, where it makes more sense: in that scenario, you have two induced actions of $k$, so it makes sense to require that they're equal (and moreover, equal to a given one).
Jan
2
comment A module over an algebra. Is it a vector space?
A small caveat. Let $M$ be a $k$-module. Suppose moreover that $M$ is an $A$-module. Then this last $A$-action induces another $k$-action that need not coincide with the original one. This reminds me of this: math.stackexchange.com/questions/889130/…
Jan
2
accepted Hochschild homology: change of ground ring
Jan
2
comment Hochschild homology: change of ground ring
Thanks a lot! I didn't know this convention: as you suspected, I was only aware of the "ring"-version of a bimodule... It's nice to learn it by making a small tedious routine verification and realizing that something was missing.
Jan
1
revised Hochschild homology: change of ground ring
edited tags
Jan
1
comment How do you break up an exact sequence of any length to a “succession of short exact sequences”?
Duplicate? math.stackexchange.com/questions/207551/…
Jan
1
asked Hochschild homology: change of ground ring