What Is the Geometric Difference in Intersection Forms Q(a,b) vs Q(b,a)?

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SUMMARY

The discussion focuses on the geometric differences between intersection forms Q(a,b) and Q(b,a) in the context of a 2n-manifold. It is established that if n is even, both forms share the same sign, while they exhibit opposite signs when n is odd. Additionally, the conversation addresses the implications of H_n being zero for a 2n-manifold M, concluding that Q must also equal zero due to bilinearity, indicating a zero net intersection despite potential actual intersections.

PREREQUISITES
  • Understanding of intersection forms in algebraic topology
  • Familiarity with homology classes and their properties
  • Knowledge of bilinear forms and their geometric interpretations
  • Concept of manifolds, specifically 2n-manifolds
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  • Research the properties of intersection forms in algebraic topology
  • Study the implications of bilinearity in homology theory
  • Explore the geometric interpretations of homology classes
  • Learn about the behavior of intersection forms in odd and even dimensions
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Mathematicians, particularly those specializing in algebraic topology, geometry enthusiasts, and students studying manifold theory will benefit from this discussion.

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Hi, everyone:

This should be easy, but I am having trouble with it. I am rusty and trying

to get back in the game:

Let Q(a,b) be an intersection form in the middle homology class

of some 2n-manifold.

What is the geometric difference between Q(a,b) and Q(b,a).?

If n is even, they are of the same sign, opposite sign

if n odd, but I am not clear on what the geometric

difference is with the different orders.


2) Also: Am I missing something really obvious here:

If H_n==0 for a 2n-manifold M . Does it follow (by bilinearity)

that Q==0.?. Since the only class is the zero class, it

would seem to follow right away. What is the geometry behind

this.?. I understand that this does not imply that there is

no actual intersection, but that the (signed) net intersection

is zero. (If above is correct) Anyone have an insight on the

geometry behind this.?

Thanks.
 

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