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Dec
8
awarded  Popular Question
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
4
asked Two-sided bar construction for algebras: $B_*(M,R^e,R)$ is quasi-isomorphic to $M\otimes_{R^e}B(R,R,R)$
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
30
revised If $p:(E,e_0) \to (X, x_0)$ is a simply connected covering space with group of covering transformations $G$ then $G \cong\pi_1(X,x_0)$.
edited title
Nov
29
awarded  Announcer
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?
Nov
9
awarded  Enlightened
Nov
9
awarded  Nice Answer
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
18
awarded  Yearling
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.
Oct
13
comment Iterating the suspension-loop adjunction in two different ways
@NajibIdrissi: done, it's here
Oct
13
revised Iterating the suspension-loop adjunction in two different ways
deleted 233 characters in body; edited tags
Oct
13
asked Iterating adjunctions in two different ways