Geometric description of a certain sphere bundle

Clash Royale CLAN TAG#URR8PPP

up vote
8
down vote

favorite

2

It appears to be a standard fact in topology that $mathbbCmathbbP^2#-mathbbCmathbbP^2$ has a structure of a $mathbbS^2$ bundle over $mathbbS^2$. Is there a nice geometric description of the projection to the sphere?

This manifold is actually a complex algebraic variety (namely a plane with one point blown up), so the question should make some sense. The best answer would probably be a meromorphic function, but I could not find one.

reference-request dg.differential-geometry

share|cite|improve this question

asked Aug 19 at 9:58

Alex Gavrilov

3,1381136

add a comment | 

up vote
8
down vote

favorite

2

It appears to be a standard fact in topology that $mathbbCmathbbP^2#-mathbbCmathbbP^2$ has a structure of a $mathbbS^2$ bundle over $mathbbS^2$. Is there a nice geometric description of the projection to the sphere?

This manifold is actually a complex algebraic variety (namely a plane with one point blown up), so the question should make some sense. The best answer would probably be a meromorphic function, but I could not find one.

reference-request dg.differential-geometry

share|cite|improve this question

asked Aug 19 at 9:58

Alex Gavrilov

3,1381136

add a comment | 

up vote
8
down vote

favorite

2

up vote
8
down vote

favorite

2

2

It appears to be a standard fact in topology that $mathbbCmathbbP^2#-mathbbCmathbbP^2$ has a structure of a $mathbbS^2$ bundle over $mathbbS^2$. Is there a nice geometric description of the projection to the sphere?

This manifold is actually a complex algebraic variety (namely a plane with one point blown up), so the question should make some sense. The best answer would probably be a meromorphic function, but I could not find one.

reference-request dg.differential-geometry

share|cite|improve this question

asked Aug 19 at 9:58

Alex Gavrilov

3,1381136

It appears to be a standard fact in topology that $mathbbCmathbbP^2#-mathbbCmathbbP^2$ has a structure of a $mathbbS^2$ bundle over $mathbbS^2$. Is there a nice geometric description of the projection to the sphere?

This manifold is actually a complex algebraic variety (namely a plane with one point blown up), so the question should make some sense. The best answer would probably be a meromorphic function, but I could not find one.

reference-request dg.differential-geometry

share|cite|improve this question

asked Aug 19 at 9:58

Alex Gavrilov

3,1381136

share|cite|improve this question

share|cite|improve this question

asked Aug 19 at 9:58

Alex Gavrilov

3,1381136

asked Aug 19 at 9:58

Alex Gavrilov

3,1381136

asked Aug 19 at 9:58

Alex Gavrilov

3,1381136

3,1381136

add a comment | 

add a comment | 

2 Answers
2

active

oldest

votes

up vote
13
down vote



accepted

Yes. If $pinmathbbCP^2$ is a point, you can consider the blowup $X_p$ of $mathbbCP^2$ at $p$ as the space of pairs $(L,q)$ such that $LsubsetmathbbCP^2$ is a line passing through $p$ and $qin L$ is any point. Now let $MsubsetmathbbCP^2$ be any line not passing through $p$. Then one can define a map $pi:X_pto M$ by letting $pi(L,q)$ be the intersection of $L$ with $M$. Then, since $M$ is a $2$-sphere (as is every line in $mathbbCP^2$), this gives a submersion of $X_p$ onto $Msimeq S^2$ whose fibers are $2$-spheres. In fact, $pi$ is a holomorphic submersion, as is easy to verify in local coordinates.

share|cite|improve this answer

answered Aug 19 at 10:10

Robert Bryant

70.6k5206305

add a comment | 

up vote
7
down vote

I would like to add another geometric description of $mathbb Cmathbb P^n #overlinemathbb Cmathbb P^n$ as sphere bundle (same works for $mathbb Cmathbb P^n # mathbb Cmathbb P^n$):

Let $S^1 to S^2n+1 to mathbb Cmathbb P^n$ be the Hopf fibration. Then
$S^1$ acts on $mathbb R^2$ by rotations and the associated vector bundle $E$ is the normal bundle of $mathbb Cmathbb P^n-1$ in $mathbb Cmathbb P^n$. Thus the total space of $E$ is diffeomorphic to $mathbb Cmathbb P^n$ with a disc removed. Hence if we glue the disc bundles of $E$ with $overline E$ (which denotes here the the bundle with the reversed orientation) along their common boundary one obtains $mathbb Cmathbb P^n# overlinemathbb Cmathbb P^n$.

Moreover one deduces from this description that the connected sum is given as the quotient $(S^2n-1times S^2)/S^1$, where $S^1$ acts on $S^2n-1$ such that it induces the Hopf fibrations and on $S^2$ by rotations. This quotient has a projection map to $S^2n-2$ with fibre $S^2$. (This is just the associated $S^2$-bundle to the Hopf fibration)

share|cite|improve this answer

edited Aug 19 at 15:05

Michael Albanese

6,81854585

answered Aug 19 at 14:27

Panagiotis Konstantis

800714

add a comment | 

Your Answer

StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "504"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: false,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);

 

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f308655%2fgeometric-description-of-a-certain-sphere-bundle%23new-answer', 'question_page');

);

Post as a guest

Name Email

2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
13
down vote



accepted

Yes. If $pinmathbbCP^2$ is a point, you can consider the blowup $X_p$ of $mathbbCP^2$ at $p$ as the space of pairs $(L,q)$ such that $LsubsetmathbbCP^2$ is a line passing through $p$ and $qin L$ is any point. Now let $MsubsetmathbbCP^2$ be any line not passing through $p$. Then one can define a map $pi:X_pto M$ by letting $pi(L,q)$ be the intersection of $L$ with $M$. Then, since $M$ is a $2$-sphere (as is every line in $mathbbCP^2$), this gives a submersion of $X_p$ onto $Msimeq S^2$ whose fibers are $2$-spheres. In fact, $pi$ is a holomorphic submersion, as is easy to verify in local coordinates.

share|cite|improve this answer

answered Aug 19 at 10:10

Robert Bryant

70.6k5206305

add a comment | 

up vote
13
down vote



accepted

Yes. If $pinmathbbCP^2$ is a point, you can consider the blowup $X_p$ of $mathbbCP^2$ at $p$ as the space of pairs $(L,q)$ such that $LsubsetmathbbCP^2$ is a line passing through $p$ and $qin L$ is any point. Now let $MsubsetmathbbCP^2$ be any line not passing through $p$. Then one can define a map $pi:X_pto M$ by letting $pi(L,q)$ be the intersection of $L$ with $M$. Then, since $M$ is a $2$-sphere (as is every line in $mathbbCP^2$), this gives a submersion of $X_p$ onto $Msimeq S^2$ whose fibers are $2$-spheres. In fact, $pi$ is a holomorphic submersion, as is easy to verify in local coordinates.

share|cite|improve this answer

answered Aug 19 at 10:10

Robert Bryant

70.6k5206305

add a comment | 

up vote
13
down vote



accepted

up vote
13
down vote



accepted

Yes. If $pinmathbbCP^2$ is a point, you can consider the blowup $X_p$ of $mathbbCP^2$ at $p$ as the space of pairs $(L,q)$ such that $LsubsetmathbbCP^2$ is a line passing through $p$ and $qin L$ is any point. Now let $MsubsetmathbbCP^2$ be any line not passing through $p$. Then one can define a map $pi:X_pto M$ by letting $pi(L,q)$ be the intersection of $L$ with $M$. Then, since $M$ is a $2$-sphere (as is every line in $mathbbCP^2$), this gives a submersion of $X_p$ onto $Msimeq S^2$ whose fibers are $2$-spheres. In fact, $pi$ is a holomorphic submersion, as is easy to verify in local coordinates.

share|cite|improve this answer

answered Aug 19 at 10:10

Robert Bryant

70.6k5206305

Yes. If $pinmathbbCP^2$ is a point, you can consider the blowup $X_p$ of $mathbbCP^2$ at $p$ as the space of pairs $(L,q)$ such that $LsubsetmathbbCP^2$ is a line passing through $p$ and $qin L$ is any point. Now let $MsubsetmathbbCP^2$ be any line not passing through $p$. Then one can define a map $pi:X_pto M$ by letting $pi(L,q)$ be the intersection of $L$ with $M$. Then, since $M$ is a $2$-sphere (as is every line in $mathbbCP^2$), this gives a submersion of $X_p$ onto $Msimeq S^2$ whose fibers are $2$-spheres. In fact, $pi$ is a holomorphic submersion, as is easy to verify in local coordinates.

share|cite|improve this answer

answered Aug 19 at 10:10

Robert Bryant

70.6k5206305

share|cite|improve this answer

share|cite|improve this answer

answered Aug 19 at 10:10

Robert Bryant

70.6k5206305

answered Aug 19 at 10:10

Robert Bryant

70.6k5206305

answered Aug 19 at 10:10

Robert Bryant

70.6k5206305

70.6k5206305

add a comment | 

add a comment | 

up vote
7
down vote

I would like to add another geometric description of $mathbb Cmathbb P^n #overlinemathbb Cmathbb P^n$ as sphere bundle (same works for $mathbb Cmathbb P^n # mathbb Cmathbb P^n$):

Let $S^1 to S^2n+1 to mathbb Cmathbb P^n$ be the Hopf fibration. Then
$S^1$ acts on $mathbb R^2$ by rotations and the associated vector bundle $E$ is the normal bundle of $mathbb Cmathbb P^n-1$ in $mathbb Cmathbb P^n$. Thus the total space of $E$ is diffeomorphic to $mathbb Cmathbb P^n$ with a disc removed. Hence if we glue the disc bundles of $E$ with $overline E$ (which denotes here the the bundle with the reversed orientation) along their common boundary one obtains $mathbb Cmathbb P^n# overlinemathbb Cmathbb P^n$.

Moreover one deduces from this description that the connected sum is given as the quotient $(S^2n-1times S^2)/S^1$, where $S^1$ acts on $S^2n-1$ such that it induces the Hopf fibrations and on $S^2$ by rotations. This quotient has a projection map to $S^2n-2$ with fibre $S^2$. (This is just the associated $S^2$-bundle to the Hopf fibration)

share|cite|improve this answer

edited Aug 19 at 15:05

Michael Albanese

6,81854585

answered Aug 19 at 14:27

Panagiotis Konstantis

800714

add a comment | 

up vote
7
down vote

I would like to add another geometric description of $mathbb Cmathbb P^n #overlinemathbb Cmathbb P^n$ as sphere bundle (same works for $mathbb Cmathbb P^n # mathbb Cmathbb P^n$):

Let $S^1 to S^2n+1 to mathbb Cmathbb P^n$ be the Hopf fibration. Then
$S^1$ acts on $mathbb R^2$ by rotations and the associated vector bundle $E$ is the normal bundle of $mathbb Cmathbb P^n-1$ in $mathbb Cmathbb P^n$. Thus the total space of $E$ is diffeomorphic to $mathbb Cmathbb P^n$ with a disc removed. Hence if we glue the disc bundles of $E$ with $overline E$ (which denotes here the the bundle with the reversed orientation) along their common boundary one obtains $mathbb Cmathbb P^n# overlinemathbb Cmathbb P^n$.

Moreover one deduces from this description that the connected sum is given as the quotient $(S^2n-1times S^2)/S^1$, where $S^1$ acts on $S^2n-1$ such that it induces the Hopf fibrations and on $S^2$ by rotations. This quotient has a projection map to $S^2n-2$ with fibre $S^2$. (This is just the associated $S^2$-bundle to the Hopf fibration)

share|cite|improve this answer

edited Aug 19 at 15:05

Michael Albanese

6,81854585

answered Aug 19 at 14:27

Panagiotis Konstantis

800714

add a comment | 

up vote
7
down vote

up vote
7
down vote

I would like to add another geometric description of $mathbb Cmathbb P^n #overlinemathbb Cmathbb P^n$ as sphere bundle (same works for $mathbb Cmathbb P^n # mathbb Cmathbb P^n$):

Let $S^1 to S^2n+1 to mathbb Cmathbb P^n$ be the Hopf fibration. Then
$S^1$ acts on $mathbb R^2$ by rotations and the associated vector bundle $E$ is the normal bundle of $mathbb Cmathbb P^n-1$ in $mathbb Cmathbb P^n$. Thus the total space of $E$ is diffeomorphic to $mathbb Cmathbb P^n$ with a disc removed. Hence if we glue the disc bundles of $E$ with $overline E$ (which denotes here the the bundle with the reversed orientation) along their common boundary one obtains $mathbb Cmathbb P^n# overlinemathbb Cmathbb P^n$.

Moreover one deduces from this description that the connected sum is given as the quotient $(S^2n-1times S^2)/S^1$, where $S^1$ acts on $S^2n-1$ such that it induces the Hopf fibrations and on $S^2$ by rotations. This quotient has a projection map to $S^2n-2$ with fibre $S^2$. (This is just the associated $S^2$-bundle to the Hopf fibration)

share|cite|improve this answer

edited Aug 19 at 15:05

Michael Albanese

6,81854585

answered Aug 19 at 14:27

Panagiotis Konstantis

800714

I would like to add another geometric description of $mathbb Cmathbb P^n #overlinemathbb Cmathbb P^n$ as sphere bundle (same works for $mathbb Cmathbb P^n # mathbb Cmathbb P^n$):

Let $S^1 to S^2n+1 to mathbb Cmathbb P^n$ be the Hopf fibration. Then
$S^1$ acts on $mathbb R^2$ by rotations and the associated vector bundle $E$ is the normal bundle of $mathbb Cmathbb P^n-1$ in $mathbb Cmathbb P^n$. Thus the total space of $E$ is diffeomorphic to $mathbb Cmathbb P^n$ with a disc removed. Hence if we glue the disc bundles of $E$ with $overline E$ (which denotes here the the bundle with the reversed orientation) along their common boundary one obtains $mathbb Cmathbb P^n# overlinemathbb Cmathbb P^n$.

Moreover one deduces from this description that the connected sum is given as the quotient $(S^2n-1times S^2)/S^1$, where $S^1$ acts on $S^2n-1$ such that it induces the Hopf fibrations and on $S^2$ by rotations. This quotient has a projection map to $S^2n-2$ with fibre $S^2$. (This is just the associated $S^2$-bundle to the Hopf fibration)

share|cite|improve this answer

edited Aug 19 at 15:05

Michael Albanese

6,81854585

answered Aug 19 at 14:27

Panagiotis Konstantis

800714

share|cite|improve this answer

share|cite|improve this answer

edited Aug 19 at 15:05

Michael Albanese

6,81854585

edited Aug 19 at 15:05

Michael Albanese

6,81854585

edited Aug 19 at 15:05

Michael Albanese

6,81854585

6,81854585

answered Aug 19 at 14:27

Panagiotis Konstantis

800714

answered Aug 19 at 14:27

Panagiotis Konstantis

800714

answered Aug 19 at 14:27

Panagiotis Konstantis

800714

800714

add a comment | 

add a comment | 

 

draft saved

draft discarded

 

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f308655%2fgeometric-description-of-a-certain-sphere-bundle%23new-answer', 'question_page');

);

Post as a guest

Name Email

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Name Email

Name

Email

Name Email

Name

Email