Fdtxjr

Instead of Universal Basic Income, why not Universal Basic NEEDS?

How difficult is it to simply disable/disengage the MCAS on Boeing 737 Max 8 & 9 Aircraft?

Can a druid choose the size of its wild shape beast?

What is a^b and (a&b)<<1?

Is a party consisting of only a bard, a cleric, and a warlock functional long-term?

Why one should not leave fingerprints on bulbs and plugs?

Employee lack of ownership

Gantt Chart like rectangles with log scale

Do the common programs (for example: "ls", "cat") in Linux and BSD come from the same source code?

Interplanetary conflict, some disease destroys the ability to understand or appreciate music

Identifying the interval from A♭ to D♯

Professor being mistaken for a grad student

Define, (actually define) the "stability" and "energy" of a compound

Python if-else code style for reduced code for rounding floats

Credit cards used everywhere in Singapore or Malaysia?

Why doesn't the EU now just force the UK to choose between referendum and no-deal?

Dice rolling probability game

Adventure Game (text based) in C++

Why do passenger jet manufacturers design their planes with stall prevention systems?

Welcoming 2019 Pi day: How to draw the letter π?

Is it true that good novels will automatically sell themselves on Amazon (and so on) and there is no need for one to waste time promoting?

What's the meaning of “spike” in the context of “adrenaline spike”?

What options are left, if Britain cannot decide?

Continuity of Linear Operator Between Hilbert Spaces

Is every 3-dimensional Poincaré complex a 3-dimensional topological variety?

Poincare duality in group (co)homologyHomotopy groups of compact topological manifoldPath or One-Dimensional ManifoldOrientability of space with only even dimensional cells.Do manifolds have the homotopy type of finite-dimensional CW-complexes?Axiomatic descrption of Poincaré polynomialQuestion about (topological) orientation of an $n$-dimensional manifoldTheorem about Poincaré algebraInduced isomorphism by Homotopy equivalenceAlexander duality on (co)chain levelsingular cohomology and Poincaré duality

0

$begingroup$

I have this question, if every 3-dimensional Poincaré complex is a 3-dimensional topological manifold?

Definition (Poincaré complex)
$X$ is a n-dimensional Poincaré complex if $X$ have the same homotopy type of a finite CW complex with a homomorphism $$ w: pi_1 (X) rightarrow mathbbZ/2 $$ and $M in H_n (X;mathbbZ[w])$, such that
$$H^ast (X;mathbbZ[w]) rightarrow ^frown [M] H_n-s(X;mathbbZ[w])$$ is a isomorphism.

algebraic-topology differential-topology homology-cohomology low-dimensional-topology poincare-duality

share|cite|improve this question

edited Mar 12 at 21:27

Erick David Luna Núñez

asked Mar 11 at 19:18

Erick David Luna NúñezErick David Luna Núñez

11

New contributor

Erick David Luna Núñez is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  By "any" do you mean "every"? And what is a "topological variety"? Do you mean "a topological manifold"?
  $endgroup$
  – Moishe Kohan
  Mar 11 at 21:31
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Welcome to Math.SE. Please review How to Ask. Especially when asking about topics from advanced subjects, Readers will reasonably expect that you give more context about the problem you want help with. Bare problem statements are often ambiguous as to where you found difficulty in approaching them, and in this way it can convey the impression that an assigned exercise is being "passed through" without the thought necessary to digest its meaning.
  $endgroup$
  – hardmath
  Mar 11 at 22:16
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  BTW, I am quite sure that the question is not an "assigned exercise", but it is poorly worded. Suggested reading: amazon.com/Elliptic-Structures-3-Manifolds-Mathematical-Society/… which contains an answer to what I think your question is.
  $endgroup$
  – Moishe Kohan
  Mar 11 at 22:46
  

  
  
  
  

add a comment | 

0

$begingroup$

I have this question, if every 3-dimensional Poincaré complex is a 3-dimensional topological manifold?

Definition (Poincaré complex)
$X$ is a n-dimensional Poincaré complex if $X$ have the same homotopy type of a finite CW complex with a homomorphism $$ w: pi_1 (X) rightarrow mathbbZ/2 $$ and $M in H_n (X;mathbbZ[w])$, such that
$$H^ast (X;mathbbZ[w]) rightarrow ^frown [M] H_n-s(X;mathbbZ[w])$$ is a isomorphism.

algebraic-topology differential-topology homology-cohomology low-dimensional-topology poincare-duality

share|cite|improve this question

edited Mar 12 at 21:27

Erick David Luna Núñez

asked Mar 11 at 19:18

Erick David Luna NúñezErick David Luna Núñez

11

New contributor

Erick David Luna Núñez is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  By "any" do you mean "every"? And what is a "topological variety"? Do you mean "a topological manifold"?
  $endgroup$
  – Moishe Kohan
  Mar 11 at 21:31
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Welcome to Math.SE. Please review How to Ask. Especially when asking about topics from advanced subjects, Readers will reasonably expect that you give more context about the problem you want help with. Bare problem statements are often ambiguous as to where you found difficulty in approaching them, and in this way it can convey the impression that an assigned exercise is being "passed through" without the thought necessary to digest its meaning.
  $endgroup$
  – hardmath
  Mar 11 at 22:16
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  BTW, I am quite sure that the question is not an "assigned exercise", but it is poorly worded. Suggested reading: amazon.com/Elliptic-Structures-3-Manifolds-Mathematical-Society/… which contains an answer to what I think your question is.
  $endgroup$
  – Moishe Kohan
  Mar 11 at 22:46
  

  
  
  
  

add a comment | 

$begingroup$

I have this question, if every 3-dimensional Poincaré complex is a 3-dimensional topological manifold?

Definition (Poincaré complex)
$X$ is a n-dimensional Poincaré complex if $X$ have the same homotopy type of a finite CW complex with a homomorphism $$ w: pi_1 (X) rightarrow mathbbZ/2 $$ and $M in H_n (X;mathbbZ[w])$, such that
$$H^ast (X;mathbbZ[w]) rightarrow ^frown [M] H_n-s(X;mathbbZ[w])$$ is a isomorphism.

algebraic-topology differential-topology homology-cohomology low-dimensional-topology poincare-duality

share|cite|improve this question

edited Mar 12 at 21:27

Erick David Luna Núñez

asked Mar 11 at 19:18

Erick David Luna NúñezErick David Luna Núñez

11

New contributor

Erick David Luna Núñez is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

edited Mar 12 at 21:27

Erick David Luna Núñez

asked Mar 11 at 19:18

Erick David Luna NúñezErick David Luna Núñez

11

New contributor

Erick David Luna Núñez is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

edited Mar 12 at 21:27

Erick David Luna Núñez

asked Mar 11 at 19:18

Erick David Luna NúñezErick David Luna Núñez

11

New contributor

Erick David Luna Núñez is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Mar 11 at 19:18

Erick David Luna NúñezErick David Luna Núñez

11

New contributor

Erick David Luna Núñez is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Mar 11 at 19:18

Erick David Luna NúñezErick David Luna Núñez

11

1 Answer
1

active

oldest

votes

1

$begingroup$

First of all, you can always modify an $n$-dimensional closed (oriented if you prefer) connected $n$-dimensional manifold to make it into a finite CW $n$-dimensional Poincare complex which is not a manifold: Just attach a 1-cell to it along a vertex of this cell. This does not change the homotopy type, hence, keeps the space $X$ a Poincare complex, but $X$ is clearly not a manifold. Thus, a meaningful question to ask is:

Is every $n$-dimensional finite Poincare complex homotopy-equivalent to an $n$-dimensional manifold?

In dimension 2, it is a nontrivial theorem due to Bieri, Linnell and Muller that the question has positive answer. In dimension 3 the answer is negative, an example is due to C.B. Thomas, it is somewhere in his book:

Elliptic Structures on 3-Manifolds, Cambridge University Press, 1986.

In fact, he had this example already in 1977 modulo the Smale Conjecture which was proven by Hatcher in 1983.

It is a famous open problem (due to C.T.C. Wall) if every finite aspherical n-dimensional Poincare complex is homotopy equivalent to an n-dimensional manifold, $nge 3$. From what I know, the question is expected to have positive answer in dimension 3 and negative answer in (sufficiently) higher dimensions. Already in dimension 3 this problem is notoriously difficult.

Two more papers to read:

C.T.C Wall, Poincare duality in dimension 3, 2004.

J. Klein, Poincare duality spaces, In: Surveys on surgery theory, Vol. 1, 135–165,
Ann. of Math. Stud., 145, Princeton Univ. Press, Princeton, NJ, 2000.

See also my answer here.

share|cite|improve this answer

edited yesterday

answered yesterday

Moishe KohanMoishe Kohan

48k344110

$endgroup$

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: "69"
;
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',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);

Erick David Luna Núñez is a new contributor. Be nice, and check out our Code of Conduct.

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

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3144105%2fis-every-3-dimensional-poincar%25c3%25a9-complex-a-3-dimensional-topological-variety%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

1

$begingroup$

First of all, you can always modify an $n$-dimensional closed (oriented if you prefer) connected $n$-dimensional manifold to make it into a finite CW $n$-dimensional Poincare complex which is not a manifold: Just attach a 1-cell to it along a vertex of this cell. This does not change the homotopy type, hence, keeps the space $X$ a Poincare complex, but $X$ is clearly not a manifold. Thus, a meaningful question to ask is:

Is every $n$-dimensional finite Poincare complex homotopy-equivalent to an $n$-dimensional manifold?

In dimension 2, it is a nontrivial theorem due to Bieri, Linnell and Muller that the question has positive answer. In dimension 3 the answer is negative, an example is due to C.B. Thomas, it is somewhere in his book:

Elliptic Structures on 3-Manifolds, Cambridge University Press, 1986.

In fact, he had this example already in 1977 modulo the Smale Conjecture which was proven by Hatcher in 1983.

It is a famous open problem (due to C.T.C. Wall) if every finite aspherical n-dimensional Poincare complex is homotopy equivalent to an n-dimensional manifold, $nge 3$. From what I know, the question is expected to have positive answer in dimension 3 and negative answer in (sufficiently) higher dimensions. Already in dimension 3 this problem is notoriously difficult.

Two more papers to read:

C.T.C Wall, Poincare duality in dimension 3, 2004.

J. Klein, Poincare duality spaces, In: Surveys on surgery theory, Vol. 1, 135–165,
Ann. of Math. Stud., 145, Princeton Univ. Press, Princeton, NJ, 2000.

See also my answer here.

share|cite|improve this answer

edited yesterday

answered yesterday

Moishe KohanMoishe Kohan

48k344110

$endgroup$

add a comment | 

1

$begingroup$

First of all, you can always modify an $n$-dimensional closed (oriented if you prefer) connected $n$-dimensional manifold to make it into a finite CW $n$-dimensional Poincare complex which is not a manifold: Just attach a 1-cell to it along a vertex of this cell. This does not change the homotopy type, hence, keeps the space $X$ a Poincare complex, but $X$ is clearly not a manifold. Thus, a meaningful question to ask is:

Is every $n$-dimensional finite Poincare complex homotopy-equivalent to an $n$-dimensional manifold?

In dimension 2, it is a nontrivial theorem due to Bieri, Linnell and Muller that the question has positive answer. In dimension 3 the answer is negative, an example is due to C.B. Thomas, it is somewhere in his book:

Elliptic Structures on 3-Manifolds, Cambridge University Press, 1986.

In fact, he had this example already in 1977 modulo the Smale Conjecture which was proven by Hatcher in 1983.

It is a famous open problem (due to C.T.C. Wall) if every finite aspherical n-dimensional Poincare complex is homotopy equivalent to an n-dimensional manifold, $nge 3$. From what I know, the question is expected to have positive answer in dimension 3 and negative answer in (sufficiently) higher dimensions. Already in dimension 3 this problem is notoriously difficult.

Two more papers to read:

C.T.C Wall, Poincare duality in dimension 3, 2004.

J. Klein, Poincare duality spaces, In: Surveys on surgery theory, Vol. 1, 135–165,
Ann. of Math. Stud., 145, Princeton Univ. Press, Princeton, NJ, 2000.

See also my answer here.

share|cite|improve this answer

edited yesterday

answered yesterday

Moishe KohanMoishe Kohan

48k344110

$endgroup$

add a comment | 

$begingroup$

First of all, you can always modify an $n$-dimensional closed (oriented if you prefer) connected $n$-dimensional manifold to make it into a finite CW $n$-dimensional Poincare complex which is not a manifold: Just attach a 1-cell to it along a vertex of this cell. This does not change the homotopy type, hence, keeps the space $X$ a Poincare complex, but $X$ is clearly not a manifold. Thus, a meaningful question to ask is:

Is every $n$-dimensional finite Poincare complex homotopy-equivalent to an $n$-dimensional manifold?

In dimension 2, it is a nontrivial theorem due to Bieri, Linnell and Muller that the question has positive answer. In dimension 3 the answer is negative, an example is due to C.B. Thomas, it is somewhere in his book:

Elliptic Structures on 3-Manifolds, Cambridge University Press, 1986.

In fact, he had this example already in 1977 modulo the Smale Conjecture which was proven by Hatcher in 1983.

It is a famous open problem (due to C.T.C. Wall) if every finite aspherical n-dimensional Poincare complex is homotopy equivalent to an n-dimensional manifold, $nge 3$. From what I know, the question is expected to have positive answer in dimension 3 and negative answer in (sufficiently) higher dimensions. Already in dimension 3 this problem is notoriously difficult.

Two more papers to read:

C.T.C Wall, Poincare duality in dimension 3, 2004.

J. Klein, Poincare duality spaces, In: Surveys on surgery theory, Vol. 1, 135–165,
Ann. of Math. Stud., 145, Princeton Univ. Press, Princeton, NJ, 2000.

See also my answer here.

share|cite|improve this answer

edited yesterday

answered yesterday

Moishe KohanMoishe Kohan

48k344110

$endgroup$

share|cite|improve this answer

edited yesterday

answered yesterday

Moishe KohanMoishe Kohan

48k344110

answered yesterday

Moishe KohanMoishe Kohan

48k344110

answered yesterday

Moishe KohanMoishe Kohan

48k344110