Manifolds homotopy equivalent to P^n # P^n

Homotopy Types of Suspended $4$-manifolds

Given a closed, smooth, connected, orientable $4$-manifold $M$, whose integral homology groups can have $2$-torsion, we determine the homotopy decomposition of the double suspension $Σ^2M$ as wedge sums of some elementary $\mathbf{A}_3^3$-complexes, which are $2$-connected finite complexes of dimension at most $6$. Furthermore, we utilize the Postnikov square (or equivalently Pontryagin square) to find sufficient conditions for the homotopy decompositions of $Σ^2M$ to desuspend to that of $ΣM$.

Menger (Nobeling) manifolds versus Hilbert cube (space) manifolds - a categorical comparison

We show that the n-homotopy category of connected (n+1)-dimensional Menger manifolds is isomorphic to the homotopy category of connected Hilbert cube manifolds whose k-dimensional homotopy groups are trivial for each k > n.

$6$-流形在$4$-流形上的环路同伦

Loop homotopy of $6$-manifolds over $4$-manifolds

本文研究了定义在单连通闭$4$-流形上秩$3$向量丛球丛全空间的$6$-流形$M$，证明其环路化后一般同伦等价于球面环路乘积，具有上同调刚性，且在Berglund意义下满足Koszul性质。

Homotopy idempotents on manifolds and Bass' conjectures

The Bass trace conjectures are placed in the setting of homotopy idempotent selfmaps of manifolds. For the strong conjecture, this is achieved via a formulation of Geoghegan. The weaker form of the conjecture is reformulated as a comparison of ordinary and L^2-Lefschetz numbers.

$6$维流形的同伦双角锥

Suspension homotopy of $6$-manifolds

本文研究了$6$维单连通闭定向流形在双角锥后的同伦分解，借此简化了$K$-群与$KO$-群的计算，并在特殊情形下细化分解以证明流形的刚性性质。

On the realization of symplectic algebras and rational homotopy types by closed symplectic manifolds

We answer a question of Oprea-Tralle on the realizability of symplectic algebras by symplectic manifolds in dimensions divisible by four, along with a question of Lupton-Oprea in all even dimensions. This will also allow us to address, in all even dimensions six and higher, another question of Oprea-Tralle on the possibility of algebraic conditions on the rational homotopy minimal model of a closed smooth manifold implying the existence of a symplectic structure on the manifold.

Higher order generalization of Fukaya's Morse homotopy invariant of 3-manifolds II. Invariants of 3-manifolds with $b_1=1$

In this paper, it is explained that a topological invariant for 3-manifold $M$ with $b_1(M)=1$ can be constructed by applying Fukaya's Morse homotopy theoretic approach for Chern--Simons perturbation theory to a local system on $M$ of rational functions associated to the free abelian covering of $M$. Our invariant takes values in Garoufalidis--Rozansky's space of Jacobi diagrams whose edges are colored by rational functions. It is expected that our invariant gives a lot of nontrivial finite type invariants of 3-manifolds.

具有基本群ℤ/p ⋊ ℤ的4维流形的同伦分类

Homotopy classification of $4$-manifolds with fundamental group $\mathbb{Z}/p \rtimes \mathbb{Z}$

本文针对奇素数p，研究具有基本群ℤ/p ⋊ ℤ的闭4维流形的同伦分类，确定其同伦型的判定依据，推广了Hambleton–Kreck的相关结果并证明了普适性分类部分结论。

开可缩流形的恰当同伦非刚性

Proper Homotopy Nonrigidity of Open Contractible Manifolds

研究开可缩流形的恰当同伦非刚性，通过特定条件构造出无限多个内部恰当同伦等价但两两非同胚的紧致可缩光滑\(N\)维流形，其边界有特定性质，揭示了与\(\mathbb{R}^n\)恰当同伦型刚性不同的情况。

The Loop Space Homotopy Type of Simply-connected Four-manifolds and their Generalizations

We determine loop space decompositions of simply-connected four-manifolds, $(n-1)$-connected $2n$-dimensional manifolds provided $n\notin\{4,8\}$, and connected sums of products of two spheres. These are obtained as special cases of a more general loop space decomposition of certain torsion-free $CW$-complexes with well-behaved skeleta and some Poincaré duality features.

高维流形的广义二重结构与简单同伦型

Generalised doubles and simple homotopy types of high dimensional manifolds

本文刻画了高维流形中存在 h-配边但非简单同伦等价例子的基本群条件，并基于加厚体二重结构及 Whitehead 挠率公式，证明当 n≥12 为偶数且 Wh(G) 对合非平凡时必存在此类例子。

悬垂单连通 $6$ 流形的同伦型

The Homotopy Types of Suspended Simply-connected $6$-manifolds

本文在 Adem 运算平凡条件下，确定了含 2-挠的单连通闭 6 流形的三重悬垂同伦型。

关于4-流形微分同胚的同伦与伪迷向的注记

A note on homotopy and pseudoisotopy of diffeomorphisms of $4$-manifolds

本文为数学领域注记，给出闭光滑4-流形上同伦于恒等映射却非伪迷向的微分同胚实例，同时阐明可定向且基本群为自由群的4-流形不存在这类例子的原因。

Rational homotopy and simply-connected 8-manifolds

We introduce a rational homotopy invariant P of a topological space, which is a quintic tensor on the cohomology. For n > 1, formality of a closed (n-1)-connected manifold of dimension up to 5n-2 is equivalent to vanishing of P and the Bianchi-Massey tensor introduced by Crowley and the second author arXiv:1505.04184. We show also that elements of the group of closed simply-connected spin 8-manifolds with the cohomology of an r-fold connected sum of S^2 x S^6 are determined up to torsion by the value of P.

Homotopy invariance of 4-manifold decompositions: connected sums

We show, up to h-cobordism, that the existence and uniqueness of connected sum decompositions of oriented 4-dimensional manifolds is an invariant of homotopy equivalence, assuming that the fundamental group of each summand is "good" in the sense of Freedman and Quinn. On a separate note, we observe that the Borel Conjecture is true in dimension 4 up to s-cobordism, assuming that the fundamental group satisfies the Farrell--Jones Conjecture.

具有有限阿贝尔2生成元基本群的4维流形的同伦分类

Homotopy classification of 4-manifolds with finite abelian 2-generator fundamental groups

本研究针对满足特定基本群条件的定向4维庞加莱复形，证明其同伦型可由二次2型确定，推进了4维流形同伦分类相关研究。

同伦等价于球面乘积的光滑流形

Smooth manifolds homotopy equivalent to products of spheres

分类了同伦等价于特定球面乘积的光滑闭定向流形，通过球面丛、Milnor 管道连接和 Novikov 球面丛等显式构造实现了正常不变映射的像。

Homotopy classification of 4-manifolds whose fundamental group is dihedral

We show that the homotopy type of a finite oriented Poincaré 4-complex is determined by its quadratic 2-type provided its fundamental group is finite and has a dihedral Sylow 2-subgroup. By combining with results of Hambleton-Kreck and Bauer, this applies in the case of smooth oriented 4-manifolds whose fundamental group is a finite subgroup of SO(3). An important class of examples are elliptic surfaces with finite fundamental group.

Homotopy properties of knots in prime manifolds

We define homotopy-theoretic invariants of knots in prime 3-manifolds. Fix a knot J in a prime 3-manifold M. Call a knot K in M concordant to J if it cobounds a properly embedded annulus with J in MxI, and call K J-characteristic if there is a degree-one map f:M --> M throwing K onto J and mapping M-K to M-J. These invariants are invariants of concordance and of J-characteristicness when f induces the identity on the fundamental group of M, and may be viewed as extensions of Milnor's invariants. We do not require the knots considered here to be rationally null-homologous or framed.

高连通庞加莱复形上的流形结构

Manifold structures on highly connected Poincaré complexes

本文构造了同伦等价于拓扑流形却不同伦等价于光滑流形的高连通庞加莱复形，并确定了k=7、15、31时特定Kervaire不变量1的闭带边流形的同伦型。

同伦与同调：5维流形中的2球面

Homotopy versus isotopy: 2-spheres in 5-manifolds

本文研究了5维流形中2球面的同伦与同调关系，通过Whitney移动消除二重点，证明了在特定条件下同伦等价于同调。

四维流形上的非动力学同伦相干作用

Non-kinetic homotopy coherent actions on four-manifolds

该研究给出闭单连通光滑四维流形上二阶非动力学光滑同伦相干作用的首个例子，通过限制离散圆周群在稳定化\(K3\)曲面上的作用获得，还构造相关扩张并展示特殊边界对合，证明了关于整同调球上自由对合光滑扩张的定理。

Higher order generalization of Fukaya's Morse homotopy invariant of 3-manifolds I. Invariants of homology 3-spheres

We give a generalization of Fukaya's Morse homotopy theoretic approach for 2-loop Chern--Simons perturbation theory to 3-valent graphs with arbitrary number of loops at least 2. We construct a sequence of invariants of integral homology 3-spheres with values in a space of 3-valent graphs (Jacobi diagrams or Feynman diagrams) by counting graphs in an integral homology 3-sphere satisfying certain condition described by a set of ordinary differential equations.

A locally hyperbolic 3-manifold that is not homotopy equivalent to any hyperbolic 3-manifold

We construct a locally hyperbolic 3-manifold $M$ such that $π_ 1(M)$ has no divisible subgroups. We then show that $M$ is not homotopy equivalent to any complete hyperbolic manifold.

The rational homotopy type of (n-1)-connected manifolds of dimension up to 5n-3

We define the Bianchi-Massey tensor of a topological space X to be a linear map from a subquotient of the fourth tensor power of H*(X). We then prove that if M is a closed (n-1)-connected manifold of dimension at most 5n-3 (and n > 1) then its rational homotopy type is determined by its cohomology algebra and Bianchi-Massey tensor, and that M is formal if and only if the Bianchi-Massey tensor vanishes. We use the Bianchi-Massey tensor to show that there are many (n-1)-connected (4n-1)-manifolds that are not formal but have no non-zero Massey products, and to present a classification of simply-…

偶维流形的简单同伦类型

Simple homotopy types of even dimensional manifolds

本文研究偶维闭流形的简单同伦类型，利用代数K理论、手术障碍映射和同伦自同构体构造了首个例子，展示了同伦等价但非简单同伦等价的流形族。

庞加莱对偶复形的稳定化与同伦回转

Stabilization of Poincaré duality complexes and homotopy gyrations

本文针对流形稳定化同伦理论的现有研究，开发新方法将其推广至庞加莱对偶复形的稳定化，系统研究了回转的同伦理论推广，给出固定庞加莱对偶复形回转同伦型的有限性判据。

关于同伦等价于 $S^{4k}$ 上 $S^{4k-1}$ 丛的拓扑流形的光滑与丛结构

On smooth and bundle structures on topological manifolds homotopy equivalent to $S^{4k-1}$-bundles over $S^{4k}$

本文验证了 $S^{4k}$ 上 $S^{4k-1}$ 丛全空间的拓扑结构集计算，并研究了其中元素能否实现为丛，确定了光滑到拓扑结构集遗忘映射的像，像中元素底层流形均为该丛。

分级融合2-范畴与4维流形的量子同伦不变量

Graded-fusion 2-categories and quantum homotopy invariants of 4-manifolds

本文引入融合2-范畴的3-群分次扩张，构造了4维流形的非平凡量子同伦不变量，推广了Douglas-Reutter不变量，且在该语境下推广了Pachner定理。

Homotopy motions of surfaces in $3$-manifolds

We introduce the concept of a homotopy motion of a subset in a manifold, and give a systematic study of homotopy motions of surfaces in closed orientable 3-manifolds. This notion arises from various natural problems in 3-manifold theory such as domination of manifold pairs, homotopical behavior of simple loops on a Heegaard surface, and monodromies of virtual branched covering surface bundles associated to a Heegaard splitting.

On open 3-manifolds proper homotopy equivalent to geometrically-simply connected polyhedra

We prove that an open 3-manifold proper homotopy equivalent to a geometrically simply connected polyhedron is simply connected at infinity, generalizing a theorem of V.Poenaru.

A separable manifold failing to have the homotopy type of a CW-complex

We show that the Prüfer surface, which is a separable non-metrizable 2-manifold, has not the homotopy type of a CW-complex. This will follow easily from J. H. C. Whitehead's result: if one has a good approximation of an arbitrary space by a CW-complex, which fails to be a homotopy equivalence, then the given space is not homotopy equivalent to a CW-complex.

Homotopy Equivalences of 3-Manifolds

Let $M$ be an oriented closed $3$-manifold. We prove that there exists a constant $A_M$, depending only on the manifold $M$, such that for every self-homotopy equivalence $f$ of $M$ there is an integer $k$ such that $1 \leq k \leq A_M$ and $f^k$ is homotopic to a homeomorphism.

Presentations of homotopy skein modules of oriented 3-manifolds

A new method to derive presentations of skein modules is developed. For the case of homotopy skein modules it will be shown how the topology of a 3-manifold is reflected in the structure of the module. The freeness problem for q-homotopy skein modules is solved, and a natural skein module related to linking numbers is computed.

On fillings of homotopy equivalent contact structures

This paper provides a topological method for filling contact structures on the connected sums of $S^2\times S^3$. Examples of nonsymplectomorphic strong fillings of homotopy equivalent contact structures with vanishing first Chern class on $\#_k S^2\times S^3$ $(k\geq2)$ are produced.

具有3流形基本群的4流形的同伦型

Homotopy types of 4-manifolds with 3-manifold fundamental groups

本文证明，对于基本群为有限表示$PD_3$-群且第一Stiefel-Whitney类匹配的4流形，其同伦型由基本群、第二同伦群、第一Kervaire类和等变交配完全决定。

Khovanov homotopy type, Burnside category, and products

In this paper, we give a new construction of a Khovanov homotopy type. We show that this construction gives a space stably homotopy equivalent to the Khovanov homotopy types constructed in [LS14a] and [HKK] and, as a corollary, that those two constructions give equivalent spaces. We show that the construction behaves well with respect to disjoint unions, connected sums and mirrors, verifying several conjectures from [LS14a]. Finally, combining these results with computations from [LS14c] and the refined s-invariant from [LS14b] we obtain new results about the slice genera of certain knots.

Homotopy classes of total foliations and bi-contact structures on three-manifolds

On every compact and orientable three-manifold, we construct total foliations (three codimension 1 foliations that are transverse at every point). This construction can be performed on any homotopy class of plane fields with vanishing Euler class. As a corollary we obtain similar results on bi-contact structures.

Lipschitz Homotopy Groups of Contact 3-Manifolds

We study contact 3-manifolds using the techniques of sub-Riemannian geometry and geometric measure theory, in particular establishing properties of their Lipschitz homotopy groups. We prove a biLipschitz version of the Theorem of Darboux: a contact $(2n+1)$-manifold endowed with a sub-Riemannian structure is locally biLipschitz equivalent to the Heisenberg group $\mathbb{H}^n$ with its \cc metric. Then each contact $(2n+1)$-manifold endowed with a sub-Riemannian structure is purely $k$-unrectifiable for $k>n$. We then extend results of Dejarnette et al. (arXiv:1109.4641 [math.FA]) and Wenger a…

K3型4流形上Dehn扭转的同伦连贯Nielsen实现问题

On the Homotopy Coherent Nielsen Realization Problem for K3-Type 4-Manifolds

利用族Seiberg-Witten理论，证明K3型4流形上沿(-2)-球面的Dehn扭转不能同伦连贯地Nielsen实现，从而给出经典Nielsen实现问题失败的新证明。

Homotopy Hyperbolic 3-Manifolds are Hyperbolic

This paper introduces a rigorous computer-assisted procedure for analyzing hyperbolic 3-manifolds. This technique is used to complete the proof of several long-standing rigidity conjectures in 3-manifold theory as well as to provide a new lower bound for the volume of a closed orientable hyperbolic 3-manifold. We prove the following result: \it\noindent Let $N$ be a closed hyperbolic 3-manifold. Then \begin{enumerate} \item[(1)] If $f\colon M \to N$ is a homotopy equivalence where $M$ is a closed irreducible 3-manifold, then $f$ is homotopic to a homeomorphism. \item[(2)] If $f,g\colon M\to N$…

A Homotopy-like Class Invariant for Sub-manifolds of Punctured Euclidean Spaces

We consider the $D$-dimensional Euclidean space, $\mathbb{R}^D$, with certain $(D-N)$-dimensional compact, closed and orientable sub-manifolds (which we call \emph{singularity manifolds} and represent by $\widetilde{\mathcal{S}}$) removed from it. We define and investigate the problem of finding a homotopy-like class invariant ($χ$-homotopy) for certain $(N-1)$-dimensional compact, closed and orientable sub-manifolds (which we call \emph{candidate manifolds} and represent by $ω$) of $\mathbb{R}^D \setminus \widetilde{\mathcal{S}}$, with special emphasis on computational aspects of the problem.…

带边4维流形的无穷同伦稳定类

Infinite homotopy stable class for 4-manifolds with boundary

该研究证明对任意奇素数q，存在无穷族稳定同胚、具有等距秩一等变相交配对且边界相同的带边拓扑4维流形，两两不存在可限制为边界同伦等价的同伦等价，丰富了4维流形同伦分类的相关结论。

Homotopy types of homeomorphism groups of noncompact 2-manifolds

Suppose M is a noncompact connected PL 2-manifold and let H(M)_0 denote the identity component of the homeomorphism group of M with the compact-open topology. In this paper we classify the homotopy type of H(M)_0 by showing that {\cal H}(M)_0 has the homotopy type of the circle if M is the plane, an open or half open annulus, or the punctured projective plane. In all other cases we show that H(M)_0 is homotopically trivial.

Homotopy and q-homotopy skein modules of 3-manifolds: an example in Algebra Situs

Algebra Situs is a branch of mathematics which has its roots in Jones' construction of his polynomial invariant of links and Drinfeld's work on quantum groups. It encompasses the theory of quantum invariants of knots and 3-manifolds, algebraic topology based on knots, operads, planar algebras, q-deformations, quantum groups, and overlaps with algebraic geometry, non-commutative geometry and statistical mechanics. Algebraic topology based on knots may be characterized as a study of properties of manifolds by considering links (submanifolds) in a manifold and their algebraic structure. The main …

Counting open negatively curved manifolds up to tangential homotopy equivalence

Under mild assumptions on a group G, we prove that the class of complete Riemannian n-manifolds of uniformly bounded negative sectional curvatures and with the fundamental groups isomorphic to G breaks into finitely many tangential homotopy types. It follows that many aspherical manifolds do not admit complete negatively curved metrics with prescribed curvature bounds.

关于3流形到5流形的浸入的拟全伦同伦

On quasi-holomorphic homotopies of immersions of 3-manifolds into 5-manifolds

提出拟全伦同伦概念扩展正则同伦，允许通过孤立奇点（交叉帽），并描述其局部形式及与全纯映射芽的联系，主要结果给出浸入像补基本群在拟全伦同伦下的变化规律。

具有 3-流形基本群的 4-流形的同伦分类

Homotopy classification of $4$-manifolds with $3$-manifold fundamental group

本文提出了基于群 $π$ 和同态 $w$ 的准则，使得具有相应基本群和定向特征的闭 4-流形可通过二次 2-型进行同伦分类，并在一大类 3-流形群上验证了该准则，推导出无限二面体基本群 4-流形的同胚分类。

Simply-connected manifolds with large homotopy stable classes

For every $k \geq 2$ and $n \geq 2$ we construct $n$ pairwise homotopically inequivalent simply-connected, closed $4k$-dimensional manifolds, all of which are stably diffeomorphic to one another. Each of these manifolds has hyperbolic intersection form and is stably parallelisable. In dimension $4$, we exhibit an analogous phenomenon for spin$^{c}$ structures on $S^2 \times S^2$. For $m\geq 1$, we also provide similar $(4m{-}1)$-connected $8m$-dimensional examples, where the number of homotopy types in a stable diffeomorphism class is related to the order of the image of the stable $J$-homomor…

Topology and homotopy of lattice isomorphic arrangements

We prove the existence of lattice isomorphic line arrangements having $π_1$-equivalent or homotopy-equivalent complements and non homeomorphic embeddings in the complex projective plane. We also provide two explicit examples, one is formed by real-complexified arrangements while the second is not.

Simply and tangentially homotopy equivalent but non-homeomorphic homogeneous manifolds

For each odd integer r greater than one and not divisible by three we give explicit examples of infinite families of simply and tangentially homotopy equivalent but pairwise non-homeomorphic closed homogeneous spaces with fundamental group isomorphic to Z/r. As an application we construct the first examples of manifolds which possess infinitely many metrics of nonnegative sectional curvature with pairwise non-homeomorphic homogeneous souls of codimension three with trivial normal bundle, such that their curvatures and the diameters of the souls are uniformly bounded. These manifolds are the fi…

Finiteness of simple homotopy type up to s-cobordism of aspherical 4-manifolds

In this paper we show that for a large class C of 4-manifolds each member of C has only finitely many simple homotopy type upto s-cobordism. This result generalizes a similar result of Hillman for certain complex surfaces. We also present a correction in the proof of Hillman's result.

与图恰当同伦等价的曲面及其德恩 - 尼尔森 - 贝尔映射

Surfaces proper homotopy equivalent to graphs and their Dehn-Nielsen-Baer maps

研究可能为无限型且边界非紧的第二可数可定向曲面与图恰当同伦等价的充要条件，探索相关映射类群间诱导映射的基本性质，以此为无限型曲面背景下德恩 - 尼尔森 - 贝尔类似物奠定基础。

Regular homotopy classes of immersions of 3-manifolds into 5-space

We give geometric formulae which enable us to detect (completely in some cases) the regular homotopy class of an immersion with trivial normal bundle of a closed oriented 3-manifold into 5-space. These are analogues of the geometric formulae for the Smale invariants due to Ekholm and the second author. As a corollary, we show that two embeddings into 5-space of a closed oriented 3-manifold with no 2-torsion in the second cohomology are regularly homotopic if and only if they have Seifert surfaces with the same signature. We also show that there exist two embeddings $F_0$ and $F_8 : T^3 \hookri…