Computing the induced map on homology from projective space
$begingroup$
Define a map from $F:P^1(Bbb C)times P^1(Bbb C) rightarrow P^2(Bbb C)$ by $((x,y), (z,w)) mapsto (xz, xw + yz, yw)$
What is the induced map on homologies?
$$H_p(Bbb{CP}^1 times Bbb{CP}^1) rightarrow H_p(Bbb{CP}^2)$$
This is supposed to be a simple example to work through but I don't know how to start. I know I need to figure out where the generators of the product are mapped and describe them with the generators of $Bbb{CP}^2$.
algebraic-topology homology-cohomology projective-space
edited Dec 18 '18 at 5:23
Michael Albanese
64.1k1599313
asked Dec 18 '18 at 4:00
Shelly BShelly B
383
$endgroup$
add a comment |
$begingroup$
Define a map from $F:P^1(Bbb C)times P^1(Bbb C) rightarrow P^2(Bbb C)$ by $((x,y), (z,w)) mapsto (xz, xw + yz, yw)$
What is the induced map on homologies?
$$H_p(Bbb{CP}^1 times Bbb{CP}^1) rightarrow H_p(Bbb{CP}^2)$$
This is supposed to be a simple example to work through but I don't know how to start. I know I need to figure out where the generators of the product are mapped and describe them with the generators of $Bbb{CP}^2$.
algebraic-topology homology-cohomology projective-space
edited Dec 18 '18 at 5:23
Michael Albanese
64.1k1599313
asked Dec 18 '18 at 4:00
Shelly BShelly B
383
$endgroup$
$begingroup$
I think we again need to know a little bit more about what you know to answer the question well. Firstly, you say you should figure out the action of $F_*$ on the generators, but what are your choices for generators of $H_2mathbb{C}P^2$ and $H_4mathbb{C}P^2$? The question is much more easily studied in cohomology, and if that is a tool available to you then a quick answer taking advantage of duality is available.
$endgroup$
– Tyrone
Dec 18 '18 at 10:49
add a comment |
$begingroup$
Define a map from $F:P^1(Bbb C)times P^1(Bbb C) rightarrow P^2(Bbb C)$ by $((x,y), (z,w)) mapsto (xz, xw + yz, yw)$
What is the induced map on homologies?
$$H_p(Bbb{CP}^1 times Bbb{CP}^1) rightarrow H_p(Bbb{CP}^2)$$
This is supposed to be a simple example to work through but I don't know how to start. I know I need to figure out where the generators of the product are mapped and describe them with the generators of $Bbb{CP}^2$.
algebraic-topology homology-cohomology projective-space
edited Dec 18 '18 at 5:23
Michael Albanese
64.1k1599313
asked Dec 18 '18 at 4:00
Shelly BShelly B
383
$endgroup$
Define a map from $F:P^1(Bbb C)times P^1(Bbb C) rightarrow P^2(Bbb C)$ by $((x,y), (z,w)) mapsto (xz, xw + yz, yw)$
What is the induced map on homologies?
$$H_p(Bbb{CP}^1 times Bbb{CP}^1) rightarrow H_p(Bbb{CP}^2)$$
This is supposed to be a simple example to work through but I don't know how to start. I know I need to figure out where the generators of the product are mapped and describe them with the generators of $Bbb{CP}^2$.
algebraic-topology homology-cohomology projective-space
algebraic-topology homology-cohomology projective-space
edited Dec 18 '18 at 5:23
Michael Albanese
64.1k1599313
asked Dec 18 '18 at 4:00
Shelly BShelly B
383
edited Dec 18 '18 at 5:23
Michael Albanese
64.1k1599313
asked Dec 18 '18 at 4:00
Shelly BShelly B
383
edited Dec 18 '18 at 5:23
Michael Albanese
64.1k1599313
edited Dec 18 '18 at 5:23
Michael Albanese
64.1k1599313
edited Dec 18 '18 at 5:23
Michael Albanese
64.1k1599313
64.1k1599313
asked Dec 18 '18 at 4:00
Shelly BShelly B
383
asked Dec 18 '18 at 4:00
Shelly BShelly B
383
asked Dec 18 '18 at 4:00
Shelly BShelly B
383
383
$begingroup$
I think we again need to know a little bit more about what you know to answer the question well. Firstly, you say you should figure out the action of $F_*$ on the generators, but what are your choices for generators of $H_2mathbb{C}P^2$ and $H_4mathbb{C}P^2$? The question is much more easily studied in cohomology, and if that is a tool available to you then a quick answer taking advantage of duality is available.
$endgroup$
– Tyrone
Dec 18 '18 at 10:49
add a comment |
$begingroup$
I think we again need to know a little bit more about what you know to answer the question well. Firstly, you say you should figure out the action of $F_*$ on the generators, but what are your choices for generators of $H_2mathbb{C}P^2$ and $H_4mathbb{C}P^2$? The question is much more easily studied in cohomology, and if that is a tool available to you then a quick answer taking advantage of duality is available.
$endgroup$
– Tyrone
Dec 18 '18 at 10:49
$begingroup$
I think we again need to know a little bit more about what you know to answer the question well. Firstly, you say you should figure out the action of $F_*$ on the generators, but what are your choices for generators of $H_2mathbb{C}P^2$ and $H_4mathbb{C}P^2$? The question is much more easily studied in cohomology, and if that is a tool available to you then a quick answer taking advantage of duality is available.
$endgroup$
– Tyrone
Dec 18 '18 at 10:49
$begingroup$
I think we again need to know a little bit more about what you know to answer the question well. Firstly, you say you should figure out the action of $F_*$ on the generators, but what are your choices for generators of $H_2mathbb{C}P^2$ and $H_4mathbb{C}P^2$? The question is much more easily studied in cohomology, and if that is a tool available to you then a quick answer taking advantage of duality is available.
$endgroup$
– Tyrone
Dec 18 '18 at 10:49
add a comment |
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$begingroup$
I think we again need to know a little bit more about what you know to answer the question well. Firstly, you say you should figure out the action of $F_*$ on the generators, but what are your choices for generators of $H_2mathbb{C}P^2$ and $H_4mathbb{C}P^2$? The question is much more easily studied in cohomology, and if that is a tool available to you then a quick answer taking advantage of duality is available.
$endgroup$
– Tyrone
Dec 18 '18 at 10:49