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 Yearling
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Jul
29
awarded  Yearling
Apr
26
comment Show $\mathbb{CP^2/CP^1}$ is not a retract of $\mathbb{CP^4/CP^1}$.
I'm confused- why doesn't Olivier's answer just answer the question. There is no nonzero element in degree 4 that squares to zero, so there's not even a nonzero map $H^*(\mathbb{C}P^2/\mathbb{C}P^1) \rightarrow H^*(\mathbb{C}P^4/\mathbb{C}P^1)$ let alone a retract.
Mar
20
awarded  Enlightened
Mar
20
awarded  Nice Answer
Feb
8
comment How can I show this is a homotopy.
If I break up my domain into two closed pieces and define continuous functions on each piece that agree on the intersection of the pieces, the result is a continuous function.
Jan
31
comment A Way to make the following “proof” of the Hairy Ball Theorem rigorous?
no. but someone thinks i am: I tried to take down the picture and they put it back up.
Dec
11
awarded  Good Answer
Dec
11
comment Andre-Quillen Homology of the cuspidal curve $k[x,y]/(x^2 - y^3)$
whoops, guess I should be more up on my math-social-media feeds...
Dec
11
comment Andre-Quillen Homology of the cuspidal curve $k[x,y]/(x^2 - y^3)$
The transitivity sequence should give a long exact sequence in homology, and you seem to be claiming that it collapses- why is the boundary zero? (probably I've confused my shifts somehow and the map is obviously zero?)
Nov
25
awarded  Nice Answer
Oct
17
awarded  Nice Answer
Jul
29
awarded  Yearling
Jun
3
awarded  Nice Question
Mar
28
awarded  Guru
Mar
22
comment Refining homotopy commutative maps of spectra to maps of E_{\infty}-ring spectra
math.uiuc.edu/~mando/papers/koandtmf.pdf
Feb
11
comment What are necessary and sufficient conditions for the product of spheres to be paralellizable?
This is false for $S^{2n} \times S^{2n}$ with $n>0$, because the Euler class of the tangent bundle is nonzero.
Dec
31
awarded  Enlightened
Dec
31
awarded  Nice Answer
Dec
9
comment Orientable Surface Covers Non-Orientable Surface
I don't know what you're allowed to use but a nice fact about the Euler characteristic is that if $E \rightarrow B$ is an $n$-sheeted covering, then the Euler characteristic of $E$ is $n$ times the Euler characteristic of $B$. If $E$ is orientable, the Euler characteristic also equals $2-2g$, if it's non-orientable the Euler characteristic is $2-g$ where $g$ is the genus. So... consider the double cover. :)
Nov
21
awarded  algebraic-topology