Is there a 4D compact smooth manifold with specific properties?

  • Thread starter lavinia
  • Start date
  • Tags
    Example
In summary, the conversation discusses the search for an example of a 4 dimensional compact smooth manifold that is orientable, can be smoothly embedded in R^8, has an odd Euler characteristic, and a second Stiefel-Whitney class of zero. However, it is determined that such an example may not exist due to the properties and restrictions of the manifold. The conversation also touches on the concept of spin structures and the Atiyah-Singer Index theorem. The conclusion is that every closed spin 4-manifold has an even Euler characteristic.
  • #1
lavinia
Science Advisor
Gold Member
3,309
694
I am looking for an example of a 4 dimensional compact smooth manifold that has the following properties

- it is orientable

- it can be smoothly embedded in R^8

- its Euler characteristic is odd

- its second Stiefel-Whitney class is zero
 
Last edited:
Physics news on Phys.org
  • #2
Cp^2
 
  • #3
afaik, the second stiefel-whitney class of CP^2 is nonvanishing (corollary 11.15 of Milnor-Stasheff).
 
  • #4
[Sorry, I confused in fact 'spin' with 'symplectic'.]

from MathWorld:

'A spin structure exists if and only if the second Stiefel-Whitney class
w2 of the tangent bundle of the manifold vanishes.'

So, it is enough to find a spin 4-manifold with b_2 odd to complete the
example. I am not an expert, but seems that there are lots of such
spaces.
 
  • #5
I am beginning to think that there is no example.

I am not sure if this argument is right but here goes.

Embed the manifold in R^8. The Stiefel-Whitney classes of the normal bundle must cancel the Stiefel-Whitney classes of the tangent bundle.

Since the tangent bundle is orientable and has zero second Stiefel-Whitney class the 4'th Stiefel Whitney class of the normal bundle must cancel the 4'th Stiefel-Whitney class of the tangent bundle and so can not be zero because the Euler characteristic of the manifold is odd.

But the Thom class of the normal bundle is zero because R^8 has no cohomology (with compact supports) except in dimension 8.
 
  • #6
Could you explain the last part of your argument please? (But the Thom class of the normal bundle is zero because R^8 has no cohomology (with compact supports) except in dimension 8.)

And also, why does [itex]w_1(\tau_M)w_3(\tau_M^{\perp})+w_3(\tau_M)w_1(\tau_M^{\perp})=0[/itex]??
 
Last edited:
  • #7
quasar987 said:
Could you explain the last part of your argument please? (But the Thom class of the normal bundle is zero because R^8 has no cohomology (with compact supports) except in dimension 8.)

And also, why does [itex]w_1(\tau_M)w_3(\tau_M^{\perp})+w_3(\tau_M)w_1(\tau_M^{\perp})=0[/itex]??
the first Whitney class of the tangent bundle is zero because the manifold is orientable. Since the sum of the normal and tangent bundles is trivial - because the tangent bundle of euclidean space is trivial - the normal bundle must also be orientable. So each term in the sum of the products of the odd whitney classes in the fourth mod 2 cohomology group is zero.

Poincare duality says that the Thom class of the normal bundle is dual to the homology class of the embedded manifold.But the embedded manifold is homologous to zero. Thus the Euler class of the normal bundle is zero and reducing mod 2, the 4'th Whitney class of the normal bundle is also zero.

If one compactifies R^8 into an 8 dimensional sphere, then one sees that the Thom class is null homologous since the 8 sphere has zero cohomology except in dimension 8.
 
Last edited:
  • #8
You may be right. Let me notice that by the Whitney embedding theorem every smooth n-manifold embeds smoothly in 2n-Euclidean space. So your second condition is superfluous.
Now, as your manifold is orientable with w2=0, it is a spin manifold. And that’s what i found in a book by Stephen Hawkins ‘himself’, a proof that every closed spin 4-manifold has even Euler characteristic - see attachment.


lavinia said:
I am beginning to think that there is no example.

I am not sure if this argument is right but here goes.

Embed the manifold in R^8. The Stiefel-Whitney classes of the normal bundle must cancel the Stiefel-Whitney classes of the tangent bundle.

Since the tangent bundle is orientable and has zero second Stiefel-Whitney class the 4'th Stiefel Whitney class of the normal bundle must cancel the 4'th Stiefel-Whitney class of the tangent bundle and so can not be zero because the Euler characteristic of the manifold is odd.

But the Thom class of the normal bundle is zero because R^8 has no cohomology (with compact supports) except in dimension 8.
 

Attachments

  • spin 4-manifolds.jpg
    spin 4-manifolds.jpg
    47.3 KB · Views: 371
  • #9
simeonsen_bg said:
You may be right. Let me notice that by the Whitney embedding theorem every smooth n-manifold embeds smoothly in 2n-Euclidean space. So your second condition is superfluous.
Now, as your manifold is orientable with w2=0, it is a spin manifold. And that’s what i found in a book by Stephen Hawkins ‘himself’, a proof that every closed spin 4-manifold has even Euler characteristic - see attachment.

Thanks. I think you can get this result without the Atiyah-Singer Index theorem. The arguments I gave show that the manifold can not be embedded in R^8. But the Whitney Embedding Theorem says that any smooth closed 4 manifold can be embedded in R^8. Therefore the Euler characteristic must be even.
 

1. What is an example in science?

An example in science is a specific instance or case that is used to demonstrate a concept, principle, or theory. It can be a physical object, a process, or a situation that is observed or manipulated in order to gain a better understanding of a scientific concept.

2. Why are examples important in science?

Examples are important in science because they help to illustrate and clarify complex ideas. They also provide evidence to support scientific theories and hypotheses, and can be used to make predictions about future events or phenomena.

3. How do scientists find examples?

Scientists find examples by conducting research, performing experiments, and gathering data. They may also observe natural phenomena in the world around them, or use models and simulations to create examples of complex systems.

4. Can examples in science be subjective?

Yes, examples in science can sometimes be subjective. This is because the interpretation of data and observations can vary depending on the individual's perspective and biases. However, scientists strive to make their examples as objective and unbiased as possible through rigorous research and experimentation.

5. How can examples in science be used in everyday life?

Examples in science can be used in everyday life to help us understand the world around us, make informed decisions, and solve practical problems. For example, we can use scientific examples to design more efficient technology, make healthier food choices, or understand the effects of climate change on our environment.

Similar threads

  • Differential Geometry
Replies
21
Views
586
  • Differential Geometry
Replies
3
Views
2K
  • Differential Geometry
Replies
7
Views
4K
  • Differential Geometry
Replies
19
Views
5K
  • Topology and Analysis
2
Replies
38
Views
4K
  • Differential Geometry
Replies
6
Views
2K
  • Differential Geometry
Replies
4
Views
2K
  • Beyond the Standard Models
Replies
7
Views
1K
  • Differential Geometry
Replies
19
Views
5K
Replies
1
Views
2K
Back
Top