arXiv subjects

Yuanxin Guan

Publications and source records attributed to Yuanxin Guan.

4 recordsLinked to original sources

arXiv2026

Equivariant Milnor map

The Milnor map is the homomorphism from the unitary bordism ring to the unoriented bordism ring, halving the dimension, that maps the unitary bordism classes of the complex Milnor hypersurfaces to the unoriented bordism classes of their real points. In this work, we propose to generalize this construction to the equivariant setup and we show the existence of such a map for the equivariant unitary groups of the circle and the cyclic group of order two. Furthermore, we relate the kernel of these Milnor maps to the magnetic unitary equivariant bordism groups of free conjugations.

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arXiv2026

On the toric lifting properties for simplicial $3$-spheres

We study the lifting problem for mod $2$ characteristic maps over simplicial $3$-spheres. Using a bad-block partition of the universal complex $X(\mathbb{Z}_2^4)$, we prove an avoidance criterion for liftability. We show that every simplicial $3$-sphere with at most $20$ vertices has the toric lifting property. We also obtain image-size and join-type results, and prove sharpness of the image-size bound in the universal-complex sense.

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arXiv2026

Non-finitely generated $(\mathbb{Z}_2)^k$-equivariant bordism ring

In 1998, Mukherjee and Sankaran posed two problems concerning the algebraic structure of the equivariant bordism ring of smooth closed $(\mathbb{Z}_2)^k$-manifolds with only isolated fixed points. One is the property of being finitely generated as a $\mathbb{Z}_2$-algebra, and the other is the existence of indecomposable elements. This paper definitively resolves both problems for the fully effective case. Specifically, let $\mathcal{Z}_*((\mathbb{Z}_2)^k)$ denote the equivariant bordism ring of smooth closed manifolds equipped with fully effective smooth $(\mathbb{Z}_2)^k$-actions having only isolated fixed points. We prove that $\mathcal{Z}_*((\mathbb{Z}_2)^k)$ is not finitely generated as a $\mathbb{Z}_2$-algebra for all $k\geqslant 3$. Moreover, the proof explicitly constructs an infinite family of indecomposable elements with unbounded degrees, thereby settling the second problem simultaneously.

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arXiv2025

Equivariant bordism classification of five-dimensional $(\mathbb{Z}_2)^3$-manifolds with isolated fixed points

Denote by $\mathcal{Z}_5((\mathbb{Z}_2)^3)$ the group, which is also a vector space over $\mathbb{Z}_2$, generated by equivariant unoriented bordism classes of all five-dimensional closed smooth manifolds with effective smooth $(\mathbb{Z}_2)^3$-actions fixing isolated points. We show that $\dim_{\mathbb{Z}_2} \mathcal{Z}_5((\mathbb{Z}_2)^3) = 77$ and determine a basis of $\mathcal{Z}_5((\mathbb{Z}_2)^3)$, each of which is explicitly chosen as the projectivization of a real vector bundle. Thus this gives a complete classification up to equivariant unoriented bordism of all five-dimensional closed smooth manifolds with effective smooth $(\mathbb{Z}_2)^3$-actions with isolated fixed points.

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