Ian Hambleton, Matthias Kreck
We establish a braid of interlocking exact sequences containing the group of homotopy self-equivalences of a smooth or topological 4-manifold. The braid is computed for manifolds whose fundamental group is finite of odd order.
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The aim of this paper is to give an $s$-cobordism classification of topological $4$-manifolds in terms of the standard invariants using the group of homotopy self-equivalences. Hambleton and Kreck constructed a braid to study the group of homotopy self-equivalences of $4$-manifolds. Using this braid together with the modified surgery theory of Kreck, we give an $s$-cobordism classification for certain $4$-manifolds with fundamental group $π$, such that cd $π\leq 2$.
Homotopy of manifolds stabilized by projective spaces
本文研究流形与射影空间连通和的同伦，提出了流形经射影空间稳定化后的环路同伦分解，并通过局部化方法追踪了特定乘积流形上外科手术的同伦影响。
Being a maximal compact subgroup of SL_nC, SU_n is a deformation retract of the former group. In this note we prove that, for sufficiently large n, there is no retraction of SL_nC to SU_n which preserves commutativity.
We classify, up to homeomorphism, all closed manifolds having the homotopy type of a connected sum of two copies of real projective n-space.
A neighborhood homotopy is an equivalence relation on spatial graphs which is generated by crossing changes on the same component and neighborhood equivalence. We give a complete classification of all 2-component spatial graphs up to neighborhood homotopy by the elementary divisor of a linking matrix with respect to the first homology group of each of the connected components. This also leads a kind of homotopy classification of 2-component handlebody-links.
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.
This is the second of a series of papers which are devoted to a comprehensive theory of maps between orbifolds. In this paper, we develop a basic machinery for studying homotopy classes of such maps. It contains two parts: (1) the construction of a set of algebraic invariants -- the homotopy groups, and (2) an analog of CW-complex theory. As a corollary of this machinery, the classical Whitehead theorem which asserts that a weak homotopy equivalence is a homotopy equivalence is extended to the orbifold category.
We study the spaces of embeddings of manifolds in a Euclidean space. More precisely we look at the homotopy fiber of the inclusion of these spaces to the spaces of immersions. As a main result we express the rational homotopy type of connected components of those embedding spaces through combinatorially defined $L_\infty$-algebras of diagrams.
Homotopy commutativity in quasitoric manifolds
本文证明了拟环面流形的环路空间同伦交换的充要条件，即底层多胞体为 $(Δ^3)^n$ 且特征矩阵属于特定类型，并构造了无限族同伦不等价的广义 Bott 流形，其中仅一半具有同伦交换性。
The homotopy theory of gauge groups has received considerable attention in recent decades. In this work, we study the homotopy theory of gauge groups over some high dimensional manifolds. To be more specific, we study gauge groups of bundles over $(n-1)$-connected closed $2n$-manifolds, the classification of which was determined by Wall and Freedman in the combinatorial category. We also investigate the gauge groups of the total manifolds of sphere bundles based on the classical work of James and Whitehead. Furthermore, other types of $2n$-manifolds are also considered. In all the cases, we sh…
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 $ΣM$.
We show that a homotopy equivalence between manifolds induces a correspondence between their spin^c-structures, even in the presence of 2-torsion. This is proved by generalizing spin^c-structures to Poincare complexes. A procedure is given for explicitly computing the correspondence under reasonable hypotheses.
Link-homotopy and self Delta-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Delta-equivalent) to a trivial link. We study link-homotopy and self Delta-equivalence on a certain component of a link with fixing the rest components, in other words, homotopy and Delta-equivalence of knots in the complement of a certain link. We show that Milnor invariants determine whether a knot in the complement of a trivial link is null-homotopic, and give a sufficient condition…
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.
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。
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.
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。
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.
Homotopy Self-Equivalences of $\mathbb{R}P^2\times S^2$ and a Braid for non-orientable $4$-Manifolds
该研究计算了RP²×S²的基于自同伦等价类群，构造了非定向4维流形的辫子正合序列，确定了Hambleton–Hillman流形的正规1型，还分析了RP²×S²的捏折映射性质。
Groups of proper homotopy equivalences of graphs and Nielsen Realization
本文研究局部有限连通图的真同伦等价群，赋予其波兰群拓扑并类比大映射类群，证明了紧子群可由某个真同伦等价图的单纯同构实现的Nielsen实现定理。
In the present paper we construct a one-to-one correspondence between the set of graph-knots and the set of homotopy classes of looped graphs. Moreover, the graph-knot and the homotopy class constructed from a given knot are related with this correspondence. This correspondence is given by a simple formula.
This paper gives a uniform-theoretic refinement of classical homotopy theory. Both cubical sets (with connections) and uniform spaces admit classes of weak equivalences, special cases of classical weak equivalences, appropriate for the respective Lipschitz and uniform settings. Cubical sets and uniform spaces admit the additional compatible structures of categories of (co)fibrant objects. A categorical equivalence between classical homotopy categories of cubical sets and spaces lifts to a full and faithful embedding from an associated Lipschitz homotopy category of cubical sets into an associa…
The Complexity of Proper Homotopy Equivalence of Graphs
该研究证明局部有限图的真同伦等价关系是博雷尔完全的，发现无限图中存在comeager等价类，并由此推出带裤子分解的非紧曲面同胚关系的类似结论。
Simple homotopy invariance of the loop coproduct
本文证明了弦拓扑中 Goresky-Hingston 环路余积在流形同伦等价下的变换公式，指出其涉及 Whitehead 挠率的迹，并揭示了该余积在简单同伦等价下的不变性及其在 2 维 TQFT 中的拓扑起源。
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.
We consider a natural question: "Is it true that each homotopy domination of a polyhedron over itself is a homotopy equivalence?" and a strongly related problem of K. Borsuk (1967): "Is it true that two ANR's homotopy dominating each other have the same homotopy type?" The answer was earlier known to be positive for manifolds (Bernstein-Ganea, 1959), $1$-dimensional polyhedra and polyhedra with polycyclic-by-finite fundamental groups (DK, 2005). Thus one may ask, if there exists a counterexample among $2$-dimensional polyhedra with soluble fundamental groups. In this paper we show that it cann…
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。
By defining combinatorial moves, we can define an equivalence relation on Gauss words called homotopy. In this paper we define a homotopy invariant of Gauss words. We use this to show that there exist Gauss words that are not homotopically equivalent to the empty Gauss word, disproving a conjecture by Turaev. In fact, we show that there are an infinite number of equivalence classes of Gauss words under homotopy.
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.
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。
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.
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.
Equivalence relations can be defined on Gauss phrases using combinatorial moves. In this paper we consider two closely related equivalence relations on Gauss phrases, homotopy and open homotopy. In particular, in each case, we define a new invariant and determine the values that it can attain.
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 on nanophrases is an equivalence relation defined using some data called a homotopy data triple. We define a product on homotopy data triples. We show that any homotopy data triple can be factorized into a product of prime homotopy data triples and this factorization is unique up to isomorphism and order. If a homotopy data triple is composite, we show that equivalence of nanophrases under the corresponding homotopy can be calculated just by using the homotopies given by its prime factors.
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.
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。
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.
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…
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.
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