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Zhenhua Liu
Zhenhua Liu
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I am trying to find a Riemannian geometrically well-understood set of generators of the oriented bordism ring, including the torsion parts. By a set of generators, I mean that the set generates the ring under multiplication and addition. The question is whether homogeneous spaces of compact Lie groups generate the oriented bordism ring.

For the unoriented bordism ring, the Dold manifolds are clearly homogeneous and they are a set of generators. For the oriented bordism ring, inspecting the list of generators of low dimensions, e.g., at Manifold Atlas, it seems that the low-dimension examples are homogeneous spaces, including the torsion ones, e.g., Wu manifold. For higher dimensional concrete examples, many Grassmannians over algebras, and even the special Lagrangian Grassmannians, etc., do not bound. Thus it makes me wonder if homogeneous spaces generate the oriented bordism ring.

All the general references I found are quite algebraic-flavored. I tried to answer the above question, but couldn't get anywhere beyond these concrete examples. I am sorry if my question is trivial, as I work in geometric measure theory, thus very far away from algebraic topology. Many thanks!

I am trying to find a Riemannian geometrically well-understood set of generators of the oriented bordism ring, including the torsion parts. By a set of generators, I mean that the set generates the ring under multiplication and addition. The question is whether homogeneous spaces of compact Lie groups generate the oriented bordism ring.

For the unoriented bordism ring, the Dold manifolds are clearly homogeneous and they are a set of generators. For the oriented bordism ring, inspecting the list of generators of low dimensions, e.g., at Manifold Atlas, it seems that the low-dimension examples are homogeneous spaces, including the torsion ones, e.g., Wu manifold. For higher dimensional concrete examples, many Grassmannians over algebras, and even the special Lagrangian Grassmannians, etc., do not bound. Thus it makes me wonder if homogeneous spaces generate the oriented bordism ring.

All the general references I found are quite algebraic-flavored. I tried to answer the above question, but couldn't get anywhere beyond these concrete examples. I am sorry if my question is trivial, as I work in geometric measure theory, thus very far away from algebraic topology. Many thanks!

I am trying to find a Riemannian geometrically well-understood set of generators of the oriented bordism ring, including the torsion parts. By a set of generators, I mean that the set generates the ring under multiplication and addition. The question is whether homogeneous spaces of compact Lie groups generate the oriented bordism ring.

For the oriented bordism ring, inspecting the list of generators of low dimensions, e.g., at Manifold Atlas, it seems that the low-dimension examples are homogeneous spaces, including the torsion ones, e.g., Wu manifold. For higher dimensional concrete examples, many Grassmannians over algebras, and even the special Lagrangian Grassmannians, etc., do not bound. Thus it makes me wonder if homogeneous spaces generate the oriented bordism ring.

All the general references I found are quite algebraic-flavored. I tried to answer the above question, but couldn't get anywhere beyond these concrete examples. I am sorry if my question is trivial, as I work in geometric measure theory, thus very far away from algebraic topology. Many thanks!

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Zhenhua Liu
Zhenhua Liu
  • 587
  • 5
  • 20

Is the oriented bordism ring generated by homogeneous spaces?

I am trying to find a Riemannian geometrically well-understood set of generators of the oriented bordism ring, including the torsion parts. By a set of generators, I mean that the set generates the ring under multiplication and addition. The question is whether homogeneous spaces of compact Lie groups generate the oriented bordism ring.

For the unoriented bordism ring, the Dold manifolds are clearly homogeneous and they are a set of generators. For the oriented bordism ring, inspecting the list of generators of low dimensions, e.g., at Manifold Atlas, it seems that the low-dimension examples are homogeneous spaces, including the torsion ones, e.g., Wu manifold. For higher dimensional concrete examples, many Grassmannians over algebras, and even the special Lagrangian Grassmannians, etc., do not bound. Thus it makes me wonder if homogeneous spaces generate the oriented bordism ring.

All the general references I found are quite algebraic-flavored. I tried to answer the above question, but couldn't get anywhere beyond these concrete examples. I am sorry if my question is trivial, as I work in geometric measure theory, thus very far away from algebraic topology. Many thanks!