Heather Johnston, Andrew Ranicki
The Bryant-Ferry-Mio-Weinberger surgery exact sequence for high-dimensional compact ANR homology manifolds is used to obtain transversality, splitting and bordism results for homology manifolds, generalizing previous work of Johnston.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We construct geometric examples of pseudomanifolds that satisfy the Witt condition for intersection homology Poincare duality with respect to certain fields but not others. We also compute the bordism theory of $K$-Witt spaces for an arbitrary field $K$, extending results of Siegel for $K=Q$.
This note corrects an error in the char(K)=2 case of the author's computation of the bordism groups of K-Witt spaces for the field K.
Looking at the cartesian product X \times X of a topological space X with itself, a natural map to be considered on that object is the involution that interchanges the coordinates, i.e. that maps (x, y) to (y, x). The so-called 'symmetric squaring construction' in Čech homology with Z/2-coefficients was introduced by Schick et al. 2007 as a map from the k-th Čech homology group of a space X to the 2k-th Čech homology group of X \times X divided by the above mentioned involution. It turns out to be a crucial construction in the proof of a parametrised Borsuk-Ulam Theorem. The symmetric squaring…
Using intersection theory in the context of Hilbert manifolds and geometric homology we show how to recover the main operations of string topology built by M. Chas and D. Sullivan. We also study and build an action of the homology of reduced Sullivan's chord diagrams on the singular homology of free loop spaces, extending previous results of R. Cohen and V. Godin and unifying part of the rich algebraic structure of string topology as an algebra over the prop of these reduced diagrams. Some of these operations are extended to spaces of maps from a sphere to a compact manifold.
The Homology of Complex Equivariant Bordism
研究阿贝尔紧致李群\(A\)及定向\(A -\)谱\(E\)，通过两种方式计算tom Dieck的同伦\(A -\)等变复配边谱\(MU_A\)的\(E -\)同调并修正错误，还计算了几何\(A -\)等变复配边谱\(mU_A\)的\(E -\)同调。
We compute the equivariant bordism of free oriented $(\mathbb{Z}/p)^n$-manifolds as a module over $Ω_*^{SO}$, when $p$ is an odd prime. We show, among others, that this module is canonically isomorphic to a direct sum of suspensions of multiple tensor products of $Ω^{SO}_*(B \mathbb{Z}/p)$, and that it is generated by products of standard lens spaces. This considerably improves previous calculations by various authors. Our approach relies on the investigation of the submodule of the Brown-Peterson homology of $B (\mathbb{Z}/p)^n$ generated by elements coming from proper subgroups of $(\mathbb{…
Let $f:V^n\looparrowright M^m$ be a smooth generic immersion. Then the set of points, that have at least $k$ preimages is an image of a (non-generic) immersion. If the manifolds $V^n$ and $M^m$ are oriented and $m-n$ is even, then the manifold of $k$-fold points is also oriented. In this paper we compute the oriented bordism class of the manifold of $k$-fold points in terms of the differential $df$, provided the tangent bundle of the manifold $M^m$ has a nowhere zero cross-section.
We define equivariant homology theories using bordism of stratifolds with a G-action, where G is a discrete group. Stratifolds are a generalization of smooth manifolds which were introduced by Kreck. He defines homology theories using bordism of suitable stratifolds. We develop the equivariant generalization of these ideas.
Nishida Relations in Bordism and Homology (extended abstract)
本文研究非定向配边中 Dyer-Lashof 运算与 Landweber-Novikov 运算之间的西田关系，并利用这些关系通过同调示性数表示覆盖流形代数 $N_*Σ$，证明基于覆盖空间运算及 D-环和 Q-环的性质。
We construct a new family of toric manifolds generating the unitary bordism ring. Each manifold in the family is the complex projectivisation of the sum of a line bundle and a trivial bundle over a complex projective space. We also construct a family of special unitary quasitoric manifolds which contains polynomial generators of the special unitary bordism ring with 2 inverted in dimensions >8. Each manifold in the latter family is obtained from an iterated complex projectivisation of a sum of line bundles by amending the complex structure to make the first Chern class vanish.
It is one of the most important facts in 4-dimensional topology that not every spherical homology class of a 4-manifold can be represented by an embedded sphere. In 1978, M. Freedman and R. Kirby showed that in the simply connected case, many of the obstructions to constructing such a sphere vanish if one modifies the ambient 4-manifold by adding products of 2-spheres, a process which is usually called stabilisation. In this paper, we extend this result to non-simply connected 4-manifolds and show how it is related to the Spin^c-bordism groups of Eilenberg-MacLane spaces.
In this paper we survey results and recent progresses on the equivariant bordism classification of 2-torus manifolds and unitary toric manifolds.
It is well known that Dold and Milnor manifolds give generators for the unoriented bordism algebra ${\frak{N}}_*$ over ${\Bbb{Z}}_2$. The purpose of this paper is to determine those Milnor manifolds which represent the same bordism classes in ${\frak{N}}_*$ as their Dold counterparts.
We restrict geometric tangential equivariant complex $T^n$-bordism to torus manifolds and provide a complete combinatorial description of the appropriate non-commutative ring. We discover, using equivariant $K$-theory characteristic numbers, that the information encoded in the oriented torus graph associated to a stably complex torus manifold completely describes its equivariant bordism class. We also consider the role of omnioriented quasitoric manifolds in this description.
The equivariant bordism classification of manifolds with group actions is an essential subject in the study of transformation groups. We are interesting in the action of 2-torus group $\mathbb{Z}_2^n$ and torus group $T^n$, and study the equivariant bordism of 2-torus manifolds and unitary toric manifolds. In this paper, we give a new description of the group $\mathcal{Z}_n(\mathbb{Z}_2^n)$ of 2-torus manifolds, and determine the dimention of $\mathcal{Z}_n(\mathbb{Z}_2^n)$ as a $\mathbb{Z}_2$-vector space. With the help of toric topology, Lü and Tan proved that the bordism groups $\mathcal{Z}…
Braidings of Self-Equivalences and Bordism
针对n≥4的闭光滑或拓扑流形，通过构造同伦高度笛卡尔正方形建立自等价空间与法配边理论环空间的关联，推广了Hambleton–Kreck的相关工作。
Equivariant bordism rigidity for toric manifolds
本文提出等变配边方法，证明环面流形同构等价于弱等变酉配边，并给出全定向拟环面流形的刚性结果，以拓扑框架检测几何与组合刚性。
Stable Cohomotopy in Codimensions Two and Three: From Algebraic Characterizations to Bordism-Theoretic Interpretations
本文从代数和几何角度研究二维和三维余维数中的稳定余余维数，给出了二维的完整特征和三维的3-原参数特征，并提供了定向流形和弦流形的 bordism 解释，推导了向量丛无处为零截面存在的充要条件。
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.
Bordism and resolution of singularities
本文通过改编奇点消解算法，证明复bordism到复导出orbifold及等变bordism的同调理论映射可分裂，进而为任意闭辛流形定义复cobordism取值的Gromov-Witten不变量，并约束Hamilton纤维丛空间拓扑。
Let $(Y,g)$ be a compact Riemannian manifold of positive scalar curvature (psc). It is well-known, due to Schoen-Yau, that any closed stable minimal hypersurface of $Y$ also admits a psc-metric. We establish an analogous result for stable minimal hypersurfaces with free boundary. Furthermore, we combine this result with tools from geometric measure theory and conformal geometry to study psc-bordism. For instance, assume $(Y_0,g_0)$ and $(Y_1,g_1)$ are closed psc-manifolds equipped with stable minimal hypersurfaces $X_0 \subset Y_0$ and $X_1\subset Y_1$. Under natural topological conditions, we…
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。
Quantitative asphericalization and Gromov's linear bordism problem
本文证明Gromov线性猜想的PL配边版本对广泛球面化群成立，并利用定量球面化方法将PL (4k-1)-流形的$L^2$ $\rho$-不变量界从超阶乘增长改进为阶乘增长。
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.
Associated with the Davis-Januszkiewicz theory of small covers, this paper deals with the theory of 2-torus manifolds from the viewpoint of equivariant bordism. We define a differential operator on the "dual" algebra of the unoriented $G_n$-representation algebra introduced by Conner and Floyd, where $G_n=(\Z_2)^n$. With the help of $G_n$-colored graphs (or mod 2 GKM graphs), we may use this differential operator to give a very simple description of tom Dieck-Kosniowski-Stong localization theorem in the setting of 2-torus manifolds. We then apply this to study the $G_n$-equivariant unoriented …
The Structure of the Spin^h Bordism Spectrum
本文证明MSpin^h上同调作为模2 Steenrod代数模的结构定理，得到其2-局部分裂，并显式计算至30次的上同调分解。
Let $M$ be an oriented manifold and let $\frak N$ be a set consisting of oriented closed manifolds of the same odd dimension. We consider the topological space $G_{\frak N, M}$ of commutative diagrams. Each commutative diagram consists of a few manifolds from $\frak N$ that are mapped to $M$ and a few one point spaces $pt$ that are each mapped to a pair of manifolds from $\frak N$. We consider the oriented bordism group $Ω_*(G_{\frak N, M})=\oplus_{i=0}^{\infty} Ω_i(G_{\frak N, M}).$ We introduce the operations $\star$ and $[\cdot, \cdot]$ on $Ω_*(G_{\frak N,M})\otimes \mathbb Q,$ that make $Ω…
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。
From bordisms of three-manifolds to domain walls between topological orders
本研究建立自旋三维流形与玻色阿贝尔拓扑序的对应关系，拓展至自旋配边与畴壁，构建对应畴壁并揭示其由有隙边界与无隙Narain边界CFT组合的结构，为拓扑序研究提供几何新视角。
This note proves that, for $F = \Bbb{R,C}$ or $\Bbb{H}$, the bordism classes of all non-bounding Grassmannian manifolds $G_k(F^{n+k})$, with $k < n$ and having real dimension $d$, constitute a linearly independent set in the unoriented bordism group ${\frak{N}}_d$ regarded as a ${\Bbb{Z}}_2$-vector space.
Quantitative bordism over acyclic groups and Cheeger-Gromov $ρ$-invariants
本文研究了acyclic群的量化bordism问题，并证明了PL(4k-1)流形的Cheeger-Gromov $ρ$-不变量存在普遍线性界，同时揭示了具有固定简单同调类型的流形的三角剖分复杂性。
The Kernel of the $\hat A$-Genus in Rational Spin Bordism is Generated by Ricci-Positive Manifolds
本文证明，在每个维度上，具有正里奇曲率度量的流形所表示的有理自旋配边类恰好张成$\hat A$-亏格的核，通过构造奇数次光滑完全交$Y_{m,\ell}$并利用多项式插值论证实现。
Let $X^{n}$ be an arbitrary oriented closed generalized $n$-manifold, $n\ge 5$. In our recent paper (Proc. Edinb. Math. Soc. (2) 63 (2020), no. 2, 597-607) we have constructed a map $t:\mathcal{N}(X^{n}) \to H^{st}_{n} ( X^{n}; \mathbb{L}^+)$ which extends the normal invariant map for the case when $X^{n}$ is a topological $n$-manifold. Here, $\mathcal{N}(X^{n})$ denotes the set of all normal bordism classes of degree one normal maps $(f,b): M^{n} \to X^{n},$ and $H^{st}_{*} ( X^{n}; \mathbb{E})$ denotes the Steenrod homology of the spectrum $\mathbb{E}$. An important nontrivial question arose…
This paper shows that the integral equivariant cohomology Chern numbers completely determine the equivariant geometric unitary bordism classes of closed unitary $G$-manifolds, which gives an affirmative answer to the conjecture posed by Guillemin--Ginzburg--Karshon in [20, Remark H.5, $\S3$, Appendix H], where $G$ is a torus. As a further application, we also obtain a satisfactory solution of [20, Question (A), $\S1.1$, Appendix H] on unitary Hamiltonian $G$-manifolds. Our key ingredients in the proof are the universal toric genus defined by Buchstaber--Panov--Ray and the Kronecker pairing of …
When can a map between manifolds be deformed away from itself? We describe a (normal bordism) obstruction which is often computable and in general much stronger than the classical primary obstruction in cohomology. In particular, it answers our question completely in a large dimension range. As an illustration we give explicit criteria in three sample settings: projections from Stiefel manifolds to Grassmannians, sphere bundle projections and maps defined on spheres. In the first example a theorem of Becker and Schultz concerning the framed bordism class of a compact Lie group plays a central …
The notion of a pseudocycle is introduced by McDuff and Salamon (J-holomorphic curves and quantum cohomology, University Lecture Series, Vol. 6, AMS (1994)) to provide a framework for defining Gromov-Witten invariants and quantum cohomology. This paper studies the bordism groups of pseudocycles, called pseudohomology groups. These are shown to satisfy the Eilenberg-Steenrod axioms. For smooth compact manifold pairs, it is proved that pseudohomology is naturally equivalent to homology. Easy examples show that this equivalence does not extend to the case of non-compact manifolds.
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。
We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex algebraic varieties, we show that the theory satisfies a projective bundle formula and $\A^1$-homotopy invariance. Moreover, we obtain transfer maps along projective morphisms.
Characteristic numbers of canonical toric manifolds and their applications
本文计算了典范胞腔流形的陈数、米尔诺数和庞特里亚金数，及其实数对应物的斯蒂夫尔-惠特尼数，探讨了其在单位ary、定向和非定向 bordism 类的组合刻画、单位ary bordism 环生成元的几何代表、典范胞腔流形的边界判定准则以及其浸入欧几里得空间的维数估计。
In this article, we classify 1-connected 8-dimensional Poincaré complexes, topological manifolds and smooth manifolds with the same homology as $S^3\times S^5$. Some questions of Escher-Ziller are also discussed.
The based loop space homology of a special family of homogeneous spaces, flag manifolds of connected compact Lie groups is studied. First, the rational homology of the based loop space on a complete flag manifold is calculated together with its Pontrjagin structure. Second, it is shown that the integral homology of the based loop space on a flag manifold is torsion free. This results in a description of the integral homology. In addition, the integral Pontrjagin structure is determined.
Bordism categories and orientations of moduli spaces
本文提出配边范畴新框架，研究模空间的可定向性与典范方向，并应用于多种几何模空间，证明可定向性并利用旗结构构造典范方向。
We show that for any set of primes $\mathcal{P}$ there exists a space $M_{\mathcal{P}}$ which is a homology and cohomology 3-manifold with coefficients in $\mathbb{Z}_{p}$ for $p\in \mathcal{ P}$ and is not a homology or cohomology 3-manifold with coefficients in $\mathbb{Z}_q$ for $q\not\in \mathcal{ P}$. Moreover, $M_{\mathcal{P}}$ is neither a homology nor cohomology 3-manifold with coefficients in $\mathbb{Z}$ or $\mathbb{Q}$.
We study and in some cases classify highly connected manifolds which admit a Riemannian metric with positive $p$-curvature. The $p$-curvature was defined and studied by the second author. It turns out that positivity of $p$-curvature could be preserved under surgeries of codimension at least $p+3$. This gives a key to reduce a geometrical classification problem to a topological one, in terms of relevant bordism groups and index theory. In particular, we classify 3-connected manifolds with positive 2-curvature in terms of the spin and string bordism groups, and by means of $α$-invariant and Wit…
An explicit (-1)^n-quadratic form over Z[Z^{2n}] representing the surgery problem E_8 x T^{2n} is obtained, for use in the Bryant-Ferry-Mio-Weinberger construction of 2n-dimensional exotic homology manifolds.
Sutured manifolds defined by Gabai are useful in the geometrical study of knots and 3-dimensional manifolds. On the other hand, homology cylinders are in an important position in the recent theory of homology cobordisms of surfaces and finite-type invariants. We study a relationship between them by focusing on sutured manifolds associated with a special class of knots which we call {\it homologically fibered knots}. Then we use invariants of homology cylinders to give applications to knot theory such as fibering obstructions, Reidemeister torsions and handle numbers of homologically fibered kn…
Handle homology of manifolds
本文通过几何方法证明了流形处理分解中链复形的∂²=0性质，并利用Cerf理论证明了处理同调的拓扑不变性。
Scalar Curvature And Transfer Maps In Spin And Spin^c Bordism
本文针对Spin^c流形及相关不变量$α^c$，证明其核是Stolz转移映射与另一类似映射的像之和，得到正标量曲率存在性的对应结果，并补充Stolz原论文细节、提供部分替代证明。
This is a survey of the current state of the question "Which closed connected manifolds of dimension $n\ge 5$ admit Riemannian metrics whose scalar curvature function is everywhere positive?" The introduction gives a brief overview of these results, while the body of the paper discusses the methods used in the proofs of these results. We mention the two flavors of topological obstructions to the existence of \pscm s: one is a consequence of the Weizenböck formula for the Dirac operator, the other is obtained by considering stable minimal hypersurfaces. We talk about geometric constructions of …
Bordisms and unbounded $KK$-theory
本文研究无界 $KK$-理论模型，通过引入 $KK$- bordism 关系定义 $KK$- bordism 组，为非交换几何中研究弱非交换几何提供框架，使 manifold 技术可直接应用于无界 $KK$-理论。
We study twisted $Spin^c$-manifolds over a paracompact Hausdorff space $X$ with a twisting $α: X \to K(\ZZ, 3)$. We introduce the topological index and the analytical index on the bordism group of $α$-twisted $Spin^c$-manifolds over $(X, α)$, taking values in topological twisted K-homology and analytical twisted K-homology respectively. The main result of this paper is to establish the equality between the topological index and the analytical index. We also define a notion of geometric twisted K-homology, whose cycles are geometric cycles of $(X, \a)$ analogous to Baum-Douglas's geometric cycl…
Bordism of flow modules and exact Lagrangians
本文为稳定标架化 Liouville 流形构造球谱上的 Donaldson-Fukaya 范畴，并建立拟同构提升的障碍理论，用于约束精确 Lagrangian 光滑结构及构造辛映射类。
We give a bordism-theoretic characterisation of those closed almost contact (2q+1)-manifolds (with q > 2) which admit a Stein fillable contact structure. Our method is to apply Eliashberg's h-principle for Stein manifolds in the setting of Kreck's modified surgery. As an application, we show that any simply connected almost contact 7-manifold with torsion free second homotopy group is Stein fillable. We also discuss the Stein fillability of exotic spheres and examine subcritical Stein fillability.
We provide a framework for the study of structured manifolds with singularities and their locally determined invariants. This generalizes factorization homology, or topological chiral homology, to the setting of singular manifolds equipped with various tangential structures. Examples of such factorization homology theories include intersection homology, compactly supported stratified mapping spaces, and Hochschild homology with coefficients. Factorization homology theories for singular manifolds are characterized by a generalization of the Eilenberg-Steenrod axioms. Using these axioms, we exte…
It is a classical important problem of differential topology by Thom; for a homology class of a compact manifold, can we realize this by a closed submanifold with no boundary? This is true if the degree of the class is smaller or equal to the half of the dimension of the outer manifold under the condition that the coefficient ring is Z_2. If the degree of the class is smaller or equal to 6 or equal to k-2 or k-1 under the condition that the coefficient ring is the integer ring where k is the dimension of the manifold, then this is also true. As a specific study, for 4-dimensional closed manifo…
We extend the theory of the universal eta-invariant to the case of relative bordism groups of manifolds with boundaries. This allows the construction of secondary descendants of the universal eta-invariant. We obtain an interpretation of Laures' f-invariant as an example of this general construction. As an aside we improve a recent result of Han-Zhang on the modularity of a certain power series of eta invariants.
On the computation of homology of type A real flag manifolds
该研究提出A型分裂实形式相关实旗流形胞腔同调系数的闭合可计算公式，借助置换码图移动演示计算过程，还算出第三、四同调群并给出至第六同调群自由部分的生成元。
Simplicial homology manifolds are proposed as an interesting class of geometric objects, more general than topological manifolds but still quite tractable, in which questions about the microstructure of space-time can be naturally formulated. Their string orientations are classified by $H^3$ with coefficients in an extension of the usual group of D-brane charges, by cobordism classes of homology three-spheres with trivial Rokhlin invariant.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。