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owlpride
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It's easy to construct maps of even degree from the three-sphere to real projective three-space. Do there exist maps of odd degree?
owlpride said:Thanks! One last question: how do you get the cup products? Or how do you know that x^3 = a? A priori x^2 or x^3 could be zero, couldn't they?
owlpride said:There's an easy degree 1 map from any orientable n-manifold to S^n: take a small open ball in the manifold and collapse its complement to a point. If you are a tiny bit more careful this map is even smooth.
mathwonk said:as with a C^infinity partition of unity. i.e. smooth "bump" function.
owlpride said:It's easy to construct maps of even degree from the three-sphere to real projective three-space. Do there exist maps of odd degree?
mathwonk said:In reference to the orientability of RP^3
A Deg 1 map from S^3 to RP(3) is a continuous map that takes points from the 3-dimensional sphere (S^3) to the 3-dimensional real projective space (RP(3)). The term "Deg 1" refers to the degree of the map, which is a measure of how many times the map wraps around the target space.
Deg 1 maps from S^3 to RP(3) are useful in topology and algebraic geometry. They help to understand the structure of the target space and can be used to classify different types of manifolds.
Some properties of a Deg 1 map from S^3 to RP(3) include continuity, differentiability, and preservation of the fundamental group. It also has a well-defined degree, which is an integer value.
A Deg 1 map from S^3 to RP(3) can be visualized as a continuous transformation of a 3-dimensional sphere onto a 3-dimensional real projective space. This can be represented graphically by mapping points from one space to the other.
Deg 1 maps from S^3 to RP(3) have applications in computer graphics, computer vision, and image processing. They can also be used in modeling and analyzing physical systems, such as fluid dynamics and electromagnetism.