Zbigniew Błaszczyk, Wacław Marzantowicz, Mahender Singh
Let $G$ be a compact Lie group. We prove that if $V$ and $W$ are orthogonal $G$-representations such that $V^G=W^G=\{0\}$, then a $G$-equivariant map $S(V) \to S(W)$ exists provided that $\dim V^H \leq \dim W^H$ for any closed subgroup $H\subseteq G$. This result is complemented by a reinterpretation in terms of divisibility of certain Euler classes when $G$ is a torus.
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Equivariant complex $K$-theory and the equivariant sphere spectrum are two of the most fundamental equivariant spectra. For an odd $p$-group, we calculate the zeroth homotopy Green functor of the localization of the equivariant sphere spectrum with respect to equivariant complex $K$-theory. Further, we calculate the zeroth homotopy Tambara functor structure in the case of odd cyclic $p$-groups.
We compute the $RO(C_2)$-graded Green functor $\underlineπ_\star L_{KU_{C_2}/(2)}S_{C_2}$.
The homotopy of the KU_G-local equivariant sphere spectrum
本研究基于已有零次结果，计算了奇素数q的有限q-群下$KU_G$-局部等变球谱的同伦Mackey函子，推进了等变同伦论相关计算。
We investigate equivariant and invariant topological complexity of spheres endowed with smooth non-free actions of cyclic groups of prime order. We prove that semilinear $\mathbb{Z}/p$-spheres have both invariants either $2$ or $3$ and calculate exact values in all but two cases for linear actions. On the other hand, we exhibit examples which show that these invariants can be arbitrarily high in the class of smooth $\mathbb{Z}/p$-spheres.
Equivariant formality of corank-one isotropy actions and products of rational spheres
本文完全刻画了满足秩差为1且左作用等变形式化的连通李群对，并修正与扩展了具有奇偶维球面乘积有理同伦型的齐性商分类。
We establish an isomorphism between the stable homotopy groups of the 2-completed motivic sphere spectrum over the real numbers and the corresponding stable homotopy groups of the 2-completed Z/2-equivariant sphere spectrum, in a certain range of dimensions.
We study otopy classes of equivariant local maps and prove the Hopf type theorem for such maps in the case of a real finite dimensional orthogonal representation of a compact Lie group.
We present an example of a homotopical localization functor which is not a localization with respect to any set of maps. Our example arises from equivariant homotopy theory. The technique of equivariant cellularization is developed and applied to the proof of the main result.
A successful generalization of phantom map theory to the equivariant case for all compact Lie groups is obtained in this paper. One of the key observations is the discovery of the fact that homotopy fiber of equivariant completion splits as product of equivariant Eilenberg-Maclane spaces which seems impossible at first sight by the example of Triantafillou.
Cohomology of spaces of Hopf equivariant maps of spheres
本文证明了球面Hopf等变映射空间的有理上同调环与相应Stiefel流形自然同构，并指出其整上同调群的自然映射是满射但非单射。
Let $G$ be a finite $p$-group. The Eilenberg-Maclane spectrum of the constant Mackey functor $\underline{\mathbb{F}}_p$, denoted $H\underline{\mathbb{F}}_p$, is modeled by the free $\mathbb{F}_p$-module on the $G$-equivariant sphere spectrum. With this construction, one has a `word length' filtration $\{(H\underline{\mathbb{F}}_p)_n\}_{n\ge 1}$. Our main theorem is that the $k$-th layer $(H\underline{\mathbb{F}}_p)_{p^k}/(H\underline{\mathbb{F}}_p)_{p^{k-1}}$ is $p$-locally equivalent to the $k$-fold suspension of the Steinberg summand of the $G$-equivariant classifying space of $(\mathbb{Z}/p…
We construct a degree-type otopy invariant for equivariant gradient local maps in the case of a real finite dimensional orthogonal representation of a compact Lie group. We prove that the invariant establishes a bijection between the set of equivariant gradient otopy classes and the direct sum of countably many copies of $\mathbb{Z}$.
We classify SO(n)-equivariant principal bundles over $S^n$ in terms of their isotropy representations over the north and south poles. This is an example of a general result classifying equivariant $(Π, G)$-bundles over cohomogeneity one manifolds.
A spectral sequence calculating the homology groups of some spaces of maps equivariant under compact group actions is described. For the main example, we calculate the rational homology groups of spaces of even and odd maps $S^m \to S^M$, $m<M$, or, which is the same, the stable homology groups of spaces of non-resultant homogeneous polynomial maps ${\mathbb R}^{m+1} \to {\mathbb R}^{M+1}$ of growing degrees. Also, we find the homology groups of spaces of ${\mathbb Z}_r$-equivariant maps of odd-dimensional spheres for any $r$. As a technical tool, we calculate the homology groups of configurat…
This paper generalizes the concept of index and co-index and some related results for free actions of G = S0 on a paracompact Hausdorff space which were introduced by Conner and Floyd. We define the index and co-index of a finitistic free G-space X, where G = Sd , d = 1 or 3 and prove that the index of X is not more than the mod 2 cohomology index of X. We observe that the index and co-index of a (2n + 1)-sphere (resp. (4n+3)-sphere) for the action of componentwise multiplication of G = S1 (resp. S3) is n. We also determine the orbit spaces of free actions of G = S3 on a finitistic space X wit…
Given a selfmap $f:X\to X$ on a compact connected polyhedron $X$, H. Schirmer gave necessary and sufficient conditions for a nonempty closed subset $A$ to be the fixed point set of a map in the homotopy class of $f$. R. Brown and C. Soderlund extended Schirmer's result to the category of fiber bundles and fiber-preserving maps. The objective of this paper is to prove an equivariant analogue of Brown-Soderlund theorem result in the category of $G$-spaces and $G$-maps where $G$ is a finite group.
Homotopical localizations with respect to a set of maps are known to exist in cofibrantly generated model categories (satisfying additional assumptions). In this paper we expand the existing framework, so that it will apply to not necessarily cofibrantly generated model categories and, more important, will allow for a localization with respect to a class of maps (satisfying some restrictive conditions). We illustrate our technique by applying it to the equivariant model category of diagrams of spaces. This model category is not cofibrantly generated. We give conditions on a class of maps which…
The filtration on the infinite symmetric product of spheres by the number of factors provides a sequence of spectra between the sphere spectrum and the integral Eilenberg-Mac Lane spectrum. This filtration has received a lot of attention and the subquotients are interesting stable homotopy types. While the symmetric product filtration has been a major focus of research since the 1980s, essentially nothing was known when one adds group actions into the picture. We investigate the equivariant stable homotopy types, for compact Lie groups, obtained from this filtration of infinite symmetric produ…
We establish a Quillen equivalence between the homotopy theories of equivariant Segal operads and equivariant simplicial operads with norm maps. Together with previous work, we further conclude that the homotopy coherent nerve is a right-Quillen equivalence from the model category of equivariant simplicial operads with norm maps to the model category structure for equivariant-$\infty$-operads in equivariant dendroidal sets.
We construct an abelian category A(G) of sheaves over a category of closed subgroups of the r-torus G and show it is of finite injective dimension. It can be used as a model for rational $G$-spectra in the sense that there is a homology theory \piA_*: G-spectra/Q --> A(G) on rational G-spectra with values in A(G), and the associated Adams spectral sequence converges for all rational $G$-spectra and collapses at a finite stage.
Equivariant "Ham Sandwich" Theorems are obtained for the finite subgroups G of the unit spheres S(F) in the classical algebras F = R, C, and H. Given any n F-valued mass distributions on F^n, it is shown that there exists a G-equivariant decomposition of F^n into |G| regular convex fundamental regions which "G-equipartition" each of the n measures, as realized by the vanishing of the "G-averages" of these regions' measures. Applications for real measures follow, among them that any n signed mass distributions on R^{(p-1)n} can be equipartitioned by a single regular p-fan when p a prime number.
A Cohomological Obstruction to the Existence of $ϕ$-Equivariant Maps
研究紧致李群间同态下\(\phi\)-等变映射存在性的障碍，通过阐述过滤群变换障碍，应用于有限循环群到圆的包含及环面间同态，为相关映射存在性研究提供新方法。
We use a well known problem in discrete and computational geometry (partitions of measures by $k$-fans) as a motivation and as a point of departure to illustrate many aspects, both theoretical and computational, of the problem of calculating the obstructions for the existence of equivariant maps. A variety of techniques are introduced and discussed with the emphasis on concrete and explicit calculations. This eventually leads (Theorems 18 and 19) to an almost exhaustive analysis of when such maps do or do not exist in the particular case of interest.
The Borel and genuine $C_2$-equivariant Adams spectral sequences
本文建立了截断实射影谱的经典 Adams 谱序列与 2-完备球面的 Borel 和 genuine C_2-等变 Adams 谱序列之间的联系，从而将 genuine 等变谱序列的计算归结为可计算的 Borel 等变谱序列。
On the degrees of equivariant maps from spheres to complex Stiefel manifolds
本文受相关学者研究启发，基于同伦论方法，利用多位学者的成果，确定了满足合适算术条件的从球面到复Stiefel流形的ℤ/m-等变映射的次数集合。
For groups of prime order, equivariant stable maps between equivariant representation spheres are investigated using the Borel cohomology Adams spectral sequence. Features of the equivariant stable homotopy category, such as stability and duality, are shown to lift to the category of modules over the associated Steenrod algebra. The dependence on the dimension functions of the representations is clarified.
A Hurewicz Theorem for $RO(C_2)$-graded Equivariant Homology Governed by Vector Fields on Spheres
本文确定了 $C_2$-等变Eilenberg-MacLane谱的 $RO(C_2)$-分次Hurewicz像，并发现其与球面上向量场问题紧密相关，进而利用广义Leibniz规则和广义Mahowald技巧证明了真 $C_2$-等变Adams谱序列中存在任意长度的非零Adams微分。
We study the homotopy types of certain spaces closely related to the spaces of algebraic (rational) maps from the $m$ dimensional real projective space into the $n$ dimensional complex projective space for $2\leq m\leq 2n$ (we conjecture this relation to be a homotopy equivalence). In an earlier article we proved that the homotopy types of the terms of the natural degree filtration approximate closer and closer the homotopy type of the space of continuous maps and obtained bounds that describe the closeness of the approximation in terms of the degrees of the maps. Here we improve the estimates…
Let \((M,ω)\) be a compact symplectic manifold of dimension \(2n\) endowed with a Hamiltonian circle action with only isolated fixed points. Whenever \(M\) admits a toric \(1\)-skeleton \(\mathcal{S}\), which is a special collection of embedded \(2\)-spheres in \(M\), we define the notion of equivariant pseudo-index of \(\mathcal{S}\): this is the minimum of the evaluation of the first Chern class \(c_1\) on the spheres of \(\mathcal{S}\). This can be seen as the analog in this category of the notion of pseudo-index for complex Fano varieties. In this paper we provide upper bounds for the equi…
Using equivariant obstruction theory we construct equivariant maps from certain classifying spaces to representation spheres for cyclic groups, product of elementary Abelian groups and dihedral groups. Restricting them to finite skeleta constructs equivariant maps between spaces which are related to the topological Tverberg conjecture. This answers negatively a question of Özaydin posed in relation to weaker versions of the same conjecture. Further, it also has consequences for Borsuk-Ulam properties of representations of cyclic and dihedral groups.
The non-equivariant topology of Stiefel manifolds has been studied extensively, culminating in a result of Miller demonstrating that a Stiefel manifold splits stably to a wedge of Thom spaces over Grassmannians. Equivariantly, one can consider spaces of isometries between representations as an analogue to Stiefel manifolds. This concept, however, yields a different theory to the non-equivariant case. We first construct a variation on the theory of the functional calculus before studying the homotopy-theoretic properties of this theory. This allows us to construct the main result; a natural tow…
Collection of (equivariant) $\rm{PL}$-mappings admitting a relative abelian, cyclic, quaternionic, bicyclic, and quaternionic-cyclic structures are constructed.
Equivariant Framed 1-Manifolds and the Pontryagin-Thom Isomorphism
本文研究等变框架1-流形与等变稳定同调群之间的同构关系，明确识别了每个元素在等变同调群中的像。
We develop a new approach to the pulling back fixed point theorem of W. Browder and use it in order to prove various generalizations of this result.
We extend the Cisinski-Moerdijk-Weiss theory of $\infty$-operads to the equivariant setting to obtain a notion of $G$-$\infty$-operads that encode "equivariant operads with norm maps" up to homotopy. At the root of this work is the identification of a suitable category of $G$-trees together with a notion of $G$-inner horns capable of encoding the compositions of norm maps. Additionally, we follow Blumberg and Hill by constructing suitable variants associated to each of the indexing systems featured in their work.
The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg-Zilber theory from Tonks' thesis, and of its relation to simplicial homotopy coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equ…
The Pontryagin-Thom theorem for families of framed equivariant manifolds
本文证明了以紧空间为参数的带标架等变流形族的庞特里亚金-托姆定理的推广，拓展了该定理的适用范围。
A trisp closure map is a special map on the vertices of a trisp T with the property that T collapses onto the subtrisp induced by the image of the map. We study the interaction between trisp closure maps and group operations on the trisp, and give conditions such that the quotient map is again a trisp closure map. Special attention is on the case that the trisp is the nerve of an acyclic category, and the relationship between trisp closure maps and closure operators on posets is studied.
We define Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure. The analogue of Froyshov's correction term in this setting is an integer-valued invariant of homology cobordism whose mod 2 reduction is the Rokhlin invariant. As an application, we show that there are no homology 3-spheres Y of Rokhlin invariant one such that Y # Y bounds an acyclic smooth 4-manifold. By previous work of Galewski-Stern and Matumoto, this implies the existence of non-triangulable high-dimensional manifolds.
We present a way of constructing and deforming diffeomorphisms of manifolds endowed with a Lie group action. This is applied to the study of exotic diffeomorphisms and involutions of spheres and to the equivariant homotopy of Lie groups.
We give an explicit decription for univeral principal U(r)-bundles with periodic map by means of equivariant Stiefel manifolds. We then show that the associated equivariant vector bundle is equivalent to the canonical one given by G. Segal. Finally, we investigate some ideals involved in the equivariant K-theory of this classifying space.
We provide a complete characterization of the equivariant commutative ring structures of all the factors in the idempotent splitting of the G-equivariant sphere spectrum, including their Hill-Hopkins-Ravenel norms, where G is any finite group. Our results describe explicitly how these structures depend on the subgroup lattice and conjugation in G. Algebraically, our analysis characterizes the multiplicative transfers on the localization of the Burnside ring of G at any idempotent element, which is of independent interest to group theorists. As an application, we obtain an explicit description …
In studying the "11/8-Conjecture" on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of Pin(2)-equivariant stable maps between certain representation spheres. In this paper, we present a complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald invariants. As a geometric application of our result, we prove a "10/8+4"-Theorem. We prove our theorem by analyzing maps between certain finite spectra arising from BPin(2) and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known re…
Let G be a finite group. For semi-free G-manifolds which are oriented in the sense of Waner, the homotopy classes of G-equivariant maps into a G-sphere are described in terms of their degrees, and the degrees occurring are characterized in terms of congruences. This is first shown to be a stable problem and then solved using methods of equivariant stable homotopy theory with respect to a semi-free G-universe.
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