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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
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
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
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
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
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/…
Dec
4
comment Verify proof that $f:X\to Y$ a homeomorphism implies $f_*:H_p(X)\to H_p(Y)$ is an isomorphism
Do you know what a functor is? Functors preserve isomorphisms. Homology is a functor.
Dec
1
comment Some questions about the Tor functor as a two-variable functor related to the arbitrary character of the choice of projective resolutions
If $A$ is a $k$-algebra and $N$ is a left $A$-module, then the bar construction $B_*(A,A,N)$ gives an $A$-resolution of $N$ as with augmentation given by the action map. Moreover, if $A$ and $N$ are $k$-flat, this is a flat resolution. This gives a nice way to compute Tor under some hypotheses.
Nov
26
comment Reference for the Universal Coefficient Spectral Sequence
Essentially, theorem 10.90 in Rotman's Intro to homological algebra, second edition.
Nov
26
comment Reference for the Universal Coefficient Spectral Sequence
mathoverflow.net/questions/165714/kunneth-spectral-sequence has a reference to Rotman. Künneth theorems are generalizations of universal coefficient ones.
Nov
17
comment why $HF_p$(Eilenberg Mac Lane spectrum) smash X (connective spectrum with finite type cohomology groups) is a wedge of suspensions of $HF_p$?
For a ring $R$, there is a Quillen equivalence between $HR$-modules and chain complexes over R. For $R=K$ a field, chain complexes split. You could think of the result you state as the spectra-side of this chain complex well-known fact.
Nov
15
comment What's the point of spectra?
Related question: math.stackexchange.com/questions/415787/…
Nov
10
comment Definition of symmetric product in Milnor's paper
Is it perhaps the join?
Oct
21
comment Homotopy groups of $S^2$
Some newer information: $\pi_n(S^2)$ is non-zero for all $n\geq 2$, see this paper
Oct
20
comment Suspension of a product - tricky homotopy equivalence
Another reference: lemma 4.2.3 in Neisendorfer's Algebraic methods in unstable homotopy theory.
Oct
13
comment Iterating the suspension-loop adjunction in two different ways
I just realized something one might say about it. If $X$ is connected, then $\Sigma \Omega \Sigma X \simeq \bigvee_n \Sigma X^{\vee n}$. Now iterate. (Hatcher proposition 4.I.2 + 4.J.1) Yikes, ok, this doesn't concatenate correctly, but I leave this here since it might be useful.