Fixed points and semifree bordism
Jeffrey D. Carlson
本文应用不动点技术计算了具有孤立不动点的半自由几何圆等变复配边的系数环，并利用19世纪方法重新推导出Sinha在2004年的结果。
我们应用不动点技术计算了具有孤立不动点的半自由几何圆等变复配边的系数环，通过19世纪的方法重新得到了Sinha在2004年的结果。
We apply fixed-point techniques to compute the coefficient ring of semifree geometric circle-equivariant complex cobordism with isolated fixed points, recovering a 2004 result of Sinha through 19th-century methods.
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A Bott tower is the total space of a tower of fibre bundles with base CP^1 and fibres CP^1. Every Bott tower of height n is a smooth projective toric variety whose moment polytope is combinatorially equivalent to an n-cube. A circle action is semifree if it is free on the complement to fixed points. We show that a (quasi)toric manifold (in the sense of Davis-Januszkiewicz) over an n-cube with a semifree circle action and isolated fixed points is a Bott tower. Then we show that every Bott tower obtained in this way is topologically trivial, that is, homeomorphic to a product of 2-spheres. This …
We show that the Hopf elements, the Kervaire classes, and the $\barκ$-family in the stable homotopy groups of spheres are detected by the Hurewicz map from the sphere spectrum to the $C_2$-fixed points of the Real Brown-Peterson spectrum. A subset of these families is detected by the $C_2$-fixed points of Real Johnson-Wilson theory $E\mathbb{R}(n)$, depending on $n$.
We study the fixed points of the universal G-equivariant n-dimensional complex vector bundle and obtain a decomposition formula in terms of twisted equivariant universal complex vector bundles of smaller dimension. We use this decomposition to describe the fixed points of the complex equivariant K-theory spectrum and the equivariant unitary bordism groups for adjacent families of subgroups.
The homotopy fixed points of Real spin bordism
该研究将spinⁿ配边的2-局部分裂细化为C₂-等变映射，推导出实自旋配边同伦不动点的2-局部分裂，并讨论了真实情形下的前景。
We give bordism-finiteness results for manifolds with semi-simple group action. Consider the class of oriented manifolds which admit a circle action with isolated fixed points such that the action extends to an $S^3$-action with fixed point. We exhibit various subclasses, characterized by an upper bound for the Euler characteristic and properties of the first Pontrjagin class, which contain only finitely many oriented bordism types in any given dimension. Also we show finiteness results for homotopy complex projective spaces and complete intersections with $S^3$-action as above.
We show that the cube of the Hopf map $η$ maps to zero under the Hurewicz map for all fixed points of all norms to cyclic $2$-groups of the Landweber-Araki Real bordism spectrum. Using that the slice spectral sequence is a spectral sequence of Mackey functors, we compute the relevant portion of the homotopy groups of these fixed points, showing that multiplication by $4$ annihilates $π_{3}$.
Equivariant bordism classification of five-dimensional $(\mathbb{Z}_2)^3$-manifolds with isolated fixed points
本文针对具有有效光滑$(\mathbb{Z}_2)^3$-作用且不动点孤立的五维闭光滑流形，确定其等变无定向配边群$\mathcal{Z}_5((\mathbb{Z}_2)^3)$的$\mathbb{Z}_2$维数为77并给出基，完成该类流形的等变无定向配边分类。
On the nonexistence of a Green functor with values spin$^c$ bordism and spin bordism
本文证明不存在以 $\mathrm{MSpin}^c$ 为底层环谱、以 $\mathrm{MSpin}$ 为 $C_2$-不动点谱的 $C_2$-环谱，从而说明相应 Green 函子不存在。
In this paper we compute homotopical bordism rings $MU^G_*$ for abelian compact Lie groups G, giving explicit generators and relations. The key constructions are operations on equivariant bordism which should play an important role in equivariant stable homotopy theory more generally. The main technique used is localization of the theory by inverting Euler classes. Applications to homotopy theory include analysis of the completion map from $MU^G_*$ to $MU^*(BG)$. Applications to geometry include classification up to cobordism of S^1 actions on stably complex four-manifolds with precisely three…
For every abelian compact Lie group A, we prove that the homotopical A-equivariant complex bordism ring, introduced by tom Dieck (1970), is isomorphic to the A-equivariant Lazard ring, introduced by Cole-Greenlees-Kriz (2000). This settles a conjecture of Greenlees. We also show an analog for homotopical real bordism rings over elementary abelian 2-groups. Our results generalize classical theorems of Quillen (1969) on the connection between non-equivariant bordism rings and formal group laws, and extend the case $A = C_2$ due to Hanke-Wiemeler (2018). We work in the framework of global homotop…
Reduced characteristic number criteria for $T^k$- and $(\mathbb Z_2)^k$-equivariant bordisms
本文为具有孤立不动点的酉T^k-流形和闭光滑(Z_2)^k-流形建立了简化的等变配边判据，分别仅依赖于单个等变陈类多项式和最高等变Stiefel-Whitney类的幂，并引入最小区分度解决了Kosniowski猜想中欧拉示性数的线性下界问题。
In this paper, we introduce the localized slice spectral sequence, a variant of the equivariant slice spectral sequence that computes geometric fixed points equipped with residue group actions. We prove convergence and recovery theorems for the localized slice spectral sequence and use it to analyze the norms of the Real bordism spectrum. As a consequence, we relate the Real bordism spectrum and its norms to a form of the $C_2$-Segal conjecture. We compute the localized slice spectral sequence of the $C_4$-norm of $BP_\mathbb{R}$ in a range and show that the Hill--Hopkins--Ravenel slice differ…
Non-finitely generated $(\mathbb{Z}_2)^k$-equivariant bordism ring
本文研究了$(\mathbb{Z}_2)^k$-equivariant bordism环的有限生成性和不可分解元素的存在性，证明对于$k\geqslant 3$，该环不是有限生成的，并构造了无限族不可分解元素。
In this work the equivariant signature of a manifold with proper action of a discrete group is defined as an invariant of equivariant bordisms. It is shown that the computation of this signature can be reduced to its computation on fixed points sets equipped with their tubular neighborhoods. It is given a description of the equivariant vector bundles with action of a discrete group $G$ for the case when the action over the base is proper quasi-free, i.e. the stationary subgroup of any point is finite. The description is given in terms of some classifying space.
We introduce a computationally tractable way to describe the $\mathbb Z$-homotopy fixed points of a $C_{n}$-spectrum $E$, producing a genuine $C_{n}$ spectrum $E^{hn\mathbb Z}$ whose fixed and homotopy fixed points agree and are the $\mathbb Z$-homotopy fixed points of $E$. These form a piece of a contravariant functor from the divisor poset of $n$ to genuine $C_{n}$-spectra, and when $E$ is an $N_{\infty}$-ring spectrum, this functor lifts to a functor of $N_{\infty}$-ring spectra. For spectra like the Real Johnson--Wilson theories or the norms of Real bordism, the slice spectral sequence pro…
Minimum numbers of fixed points or of coincidence components (realized by maps in given homotopy classes) are the principal objects of study in topological fixed point and coincidence theory. In this paper we investigate fiberwise analoga and represent a general approach e.g. to the question when two maps can be deformed until they are coincidence free. Our method involves normal bordism theory, a certain pathspace EB and a natural generalization of Nielsen numbers. As an illustration we determine the minimum numbers for all maps between torus bundles of arbitrary (possibly different) dimensio…
Real spin bordism and orientations of topological $\mathrm{K}$-theory
本文构造了交换正交 $C_2$-环谱 $\mathrm{MSpin}^c_{\mathbb{R}}$ 及 Atiyah 实 K-理论的定向，借此统一恢复拓扑 K-理论的三个经典定向，并证明 $\hat{A}$-亏格的整性阻碍了不动点与 $\mathrm{MSpin}$ 的等价。
Let M to B, N to B be fibrations and f1,f2 :M to N be a pair of fibre-preserving maps. Using normal bordism techniques we define an invariant which is an obstruction to deforming the pair f1,f2 over B to a coincidence free pair of maps.In the special case where the two fibrations are the same and one of the maps is the identity, a weak version of our ω-invariant turns out to equal Dold's fixed point index of fibre-preserving maps. The concepts of Reidemeister classes and Nielsen coincidence classes over B are developed. As an illustration we compute e.g. the minimal number of coincidence compo…
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