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Mar
7
comment Poincare Duality Reference
Is there any reference for the proof that the dual decomposition is really a CW-decomposition? Apparently, it depends on the fact that the original simplicial complex is a manifold. I don't know how to take advantage of this homogeneity explicitly.
Mar
7
comment Poincare Duality Reference
In which section of Schubert's book? I didn't find that.
Feb
24
revised Convergence in measure and convergence of norm implies convergence in L^p
added 30 characters in body
Feb
24
answered Convergence in measure and convergence of norm implies convergence in L^p
Feb
24
comment Commutative ring where $r$, $s$ are associates but $r \neq us$ for any $u$ unit.
Consider the ring $A=\mathbb Z[X,Y,Z]/(XYZ-X)=\mathbb Z[x,y,z]$, where $x,y,z$ are residues of $X,Y,Z$ in $A$.
Feb
24
asked Long exact sequence for a triple follows from long exact sequence for a pair?
Feb
23
awarded  Nice Question
Feb
23
comment If $G$ is an uncountable group and $H$ is a subgroup then $G$ \ $H$ is uncountable ?
But different $gH$'s are disjoint, therefore $G\setminus H$ is uncountable.
Feb
23
revised A short exact sequence of chain complexes with null-homotopic chain maps
added 17 characters in body
Feb
23
comment A short exact sequence of chain complexes with null-homotopic chain maps
@Hanno Sorry, $f$ is not null-homotopic, but a homotopy equivalence. Thanks!
Feb
23
revised A short exact sequence of chain complexes with null-homotopic chain maps
edited title
Feb
23
comment If $G$ is an uncountable group and $H$ is a subgroup then $G$ \ $H$ is uncountable ?
Is it the set difference? Then you should mention that $gH$ is of the same cardinality as $H$ for $g\in G$.
Feb
23
asked A short exact sequence of chain complexes with null-homotopic chain maps
Feb
23
comment acyclic implies identity null-homotopic?
Well, if you are still here, you can check Weibel, Exercise 1.4.1 that acyclic bounded below chain complexes of free (in fact, projective) $R$-modules are always split exact, and an acyclic chain complex of free abelian groups (in fact, modules over PID) is always split exact.
Feb
18
comment punctured Mobius band in high dimension
@Carl And is it true that punctured Möbius band is homotopy equivalent to $S^1$? If the code for fundamental polygon of the Möbius band is $abac$, it seems to me that the punctured Möbius band has deformation retraction to the boundary, which is 8-shaped space, not a circle.
Feb
18
comment Constructing chain homotopy equivalence related to mapping cones
@ZhenLin Maybe I was too lazy. Through a blind-search work (by guessing the grading, etc), I got a solution, but the geometric meaning is still unclear to me. For example, if $f$ corresponds to the embedding $S^1\to\overline{D^2}$, I cannot figure out where $s$ arise from geometrically.
Feb
18
revised Constructing chain homotopy equivalence related to mapping cones
added 371 characters in body
Feb
16
asked Constructing chain homotopy equivalence related to mapping cones
Feb
13
comment Visualising algebraic topology
An old-fashioned textbook: Seifert and Threlfall.
Feb
12
comment Generalizing a statement about direct limits in the category of $A$-modules to other categories
@MartinBrandenburg Now I read from Ravi Vakil's lecture notes, Jan 29 2015 draft 1.4.8. Summary that, in a category where the objects are set-like, an element of a colimit can be thought of an element of a single object in the diagram. How can we formalize set-likeness? It seems to me that it's a compatibility between limit/colimit and forgetful functor to sets. Is there any generalization?