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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?
Feb
12
accepted Book suggestions on projective geometry
Feb
10
comment Book suggestions on projective geometry
Seems interesting. The post was too archaic, so I need time to check it.
Feb
6
comment Does this sequence of functions converge uniformly on $\mathbb{R}$?
Consider $f_n(x)=0$ for $\lvert x\rvert\le n$ and $1$ otherwise.
Feb
6
comment Corverting topological problems to algebraic problems
@irem I hope that Seifert and Threlfall's A Textbook of Topology makes a good exposition to you, and Bott and Tu's Differential Forms in Algebraic Topology. Personally I think Hatcher is somewhat wordy.
Feb
6
comment Differentiability of f(x)=x ($\sqrt{x}+\sqrt{x+9}$).
@BigBang It should be one-sided differentiable. I've mentioned, though.
Feb
6
comment Differentiability of f(x)=x ($\sqrt{x}+\sqrt{x+9}$).
What you derive is $f'(x)$ for $x>0$. If you want to determine the right derivative of $f$ at $0$, you can either determine $\lim_{x\to0+}f(x)/x$ directly, or appeal to the theorem that that $\lim_{x\to0+}f'(x)$ exists and $f$ is right continuous at $x=0$ implies that $f_+'(0)$ exists.
Feb
6
revised The generic fiber of a morphism of schemes
added 26 characters in body
Feb
6
comment Let $A, B$ and $C$ be $n × n$ matrices such that $ABC = I_n$.
You should know that if $AB=I_n$ for $A,B\in k^{n\times n}$, then $BA=I_n$.
Feb
6
comment does this real sequence of integrals converge?
$\{f_n\}$ should be uniformly bounded by some $M$, therefore $\int_{1-1/n}^1\lvert f_n\rvert\le M/n$.
Feb
5
asked Two different notions of covering homotopy?
Feb
4
revised A form of Künneth formula?
added 18 characters in body
Feb
4
revised A form of Künneth formula?
added 72 characters in body
Feb
4
comment A form of Künneth formula?
@QiaochuYuan Integral.