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Dec
31
comment Contracting a contractible set in $\mathbb R^2$
@M314 Complement to the horned ball is not simply connected. So in $\mathbb R^3/A$ we have a point $A/A$ with non-simply-connected complement.
Dec
30
comment Mapping cylinder is a CW complex
Have you seen section 'Topology of Cell Complexes' in Hatcher? At least product of CW-complexes is discussed in detail there.
Dec
30
comment Cohomology Ring of Projective Space
Related: Proving that the cohomology ring of $\mathbb{R}P^n$ is isomorphic to...
Dec
30
comment Cohomology ring of $\mathbb R P^\infty$ with $\mathbb Z_{2k}$ coefficients
Related: the same question for $k=2$
Dec
30
comment Contracting a contractible set in $\mathbb R^2$
I'm not even sure if this is true — for example for $\mathbb R^3$ the statement seems to be false (take $A$ to be the horned ball...).
Dec
29
comment Contracting a contractible set in $\mathbb R^2$
$S^1\subset\mathbb R^2$ is compact and connected but not contractible.
Dec
29
comment Concrete non trivial computation of Morse homology
[Let's wait for answers but] I've always thought it's mainly useful for (co)homology of loop space and such [and not in finite-dimensional situation]...
Dec
27
comment Applications of universal coefficient theorem
possible duplicate of Why homology with coefficients?
Dec
27
comment Conditions for a Topological space to be a Spectrum
@UserSomeNumber Why, of course everything can (and should) be considered up to (coherent) homotopy equivalence
Dec
26
comment Relationship between topological and Quillen's K-theory
(Somewhat) related: Topological vs. Algebraic K-Theory
Dec
26
comment Conditions for a Topological space to be a Spectrum
@UserSomeNumber One space can have different structures of infinite loop space. So ($\Omega^\infty$-)spectrum is not just a space that can be infinitely delooped, it's a space together with some choice of such deloopings.
Dec
25
comment Intuition of Chern-Weil theory
Good question! Basically, Chern–Weil formulas come from (Lie algebra) cohomology of $g$ and its close relative $(S^\ast g[2])^{g}$ as an algebraic model for the universal bundle $EG\to BG$. (I hope to write an answer later.)
Dec
25
comment “Why” is $[\mathbb{C}:\mathbb{R}] < \infty$?
@omar That's more or less the reply I was typing :-) (Related question @ MO: mathoverflow.net/q/8756 )
Dec
25
comment “Why” is $[\mathbb{C}:\mathbb{R}] < \infty$?
For one thing, any polynomial of odd degree has a real root...
Dec
24
comment Solving 5th degree or higher equations
One way to prove this, btw, is Lagrange inversion formula.
Dec
24
comment Does a four-variable analog of the Hall-Witt identity exist?
Related: negative answer @ MO
Dec
24
comment Does a four-variable analog of the Hall-Witt identity exist?
See also lamington.wordpress.com/2011/11/20/the-hall-witt-identity
Dec
23
comment The close form expression of a Pfaffian
$\operatorname{Pf}(1/(x_j-x_i))$ can be computed using Cauchy determinant identity, perhaps
Dec
22
comment Lie algebra of Derivations as a functor?
$\operatorname{Der}(A,M)$ ($M$-valued derivations) is a functor in $M$. As for $\operatorname{Der}(A,A)$, it's in a sense an algebraic version of tangent bundle...
Dec
20
comment Triple Euler sum result $\sum_{k\geq 1}\frac{H_k^{(2)}H_k }{k^2}=\zeta(2)\zeta(3)+\zeta(5)$
Related: math.stackexchange.com/q/606070