Donald M Davis, Stephen D Theriault
We note that a recent result of the second author yields upper bounds for odd-primary homotopy exponents of compact simple Lie groups which are often quite close to the lower bounds obtained from v_1-periodic homotopy theory.
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
The $p$-local homotopy types of gauge groups of principal $G$-bundles over $S^4$ are classified when $G$ is a compact connected exceptional Lie group without $p$-torsion in homology except for $(G,p)=(\mathrm{E}_7,5)$.
In this article we study thick ideals defined by periodic self maps in the stable motivic homotopy category over $\mathbb{C}$. In addition, we extend some results of Ruth Joachimi about the relation between thick ideals defined by motivic Morava K-theories and the preimages of the thick ideals in the stable homotopy category under Betti realization.
For primes p>=3, Cohen, Moore, and Neisendorfer showed that the exponent of the p-torsion in the homotopy groups of S^2n+1 is p^n. This was obtained as a consequence of a thorough analysis of the homotopy theory of Moore spaces. Anick further developed this for p>=5 by constructing a homotopy fibration S^2n-1 --> T^2n+1(p^r) --> Loop S^2n+1 whose connecting map is degree p^r on the bottom cell. A much simpler construction of such a fibration for p>=3 was given by Gray and the author using new methods. In this paper the new methods are used to start over, first constructing Anick's fibration fo…
We determine the v1-periodic homotopy groups of all irreducible p-compact groups (BX,X). In the most difficult, modular, cases, we follow a direct path from their associated invariant polynomials to these homotopy groups. We show that, if p is odd, every irreducible p-compact group has X of the homotopy type of a product of explicit spaces related to p-completed Lie groups.
We show that H-spaces with finitely generated cohomology, as an algebra or as an algebra over the Steenrod algebra, have homotopy exponents at all primes. This provides a positive answer to a question of Stanley.
We study the asymptotic behavior of the homotopy groups of simply connected finite $p$-local complexes, and define a space to be locally hyperbolic if its homotopy groups have exponential growth. Under some certain conditions related to the functorial decomposition of loop suspension, we prove that the suspended finite complexes are locally hyperbolic if suitable but accessible information of the homotopy groups is known. In particular, we prove that Moore spaces are locally hyperbolic, and other candidates are also given.
We study connected mod p finite A_p-spaces admitting AC_n-space structures with n<p for an odd prime p. Our result shows that if n is greator than (p-1)/2, then the mod p Steenrod algebra acts on the mod p cohomology of such a space in a systematic way. Moreover, we consider A_p-spaces which are mod p homotopy equivalent to product spaces of odd dimensional spheres. Then we determine the largest integer n for which such a space admits an AC_n-space structure compatible with the A_p-space structure.
Homotopy exponents of polyhedral products
本文研究了棱柱积的同伦指数与Moore猜想，证明了在特定条件下，棱柱积在任何奇素数处均无同伦指数，并给出了判断棱柱积在有理范畴下超几何的准则。
We compute the 3-primary v1-periodic homotopy groups of the exceptional Lie group E7. Now E8 at the primes 3 and 5 is the only compact simple Lie group whose odd-primary v1-periodic homotopy groups remian to be computed. The main work is computing the unstable Novikov spectral sequence of ΩE7/Sp(2). Showing that this converges to v1-periodic homotopy groups requires recent work of Bousfield and Bendersky-Thompson.
Let $p$ be an odd prime, and fix integers $m$ and $n$ such that $0<m<n\leq (p-1)(p-2)$. We give a $p$-local homotopy decomposition for the loop space of the complex Stiefel manifold $W_{n,m}$. Similar decompositions are given for the loop space of the real and symplectic Stiefel manifolds. As an application of these decompositions, we compute upper bounds for the $p$-exponent of $W_{n,m}$. Upper bounds for $p$-exponents in the stable range $2m<n$ and $0<m\leq (p-1)(p-2)$ are computed as well.
Exponential growth in the rational homology of free loop spaces and in torsion homotopy groups
本文利用积分方法推广了自由环路空间有理同调群增长的结果，构造出同伦群$p$-挠指数增长的新空间族，并提出有理同伦群与$p$-挠同伦群指数增长间存在紧密联系的猜想。
In this article, we study the exponents of metastable homotopy of mod $2$ Moore spaces. Our result gives that the double loop space of $4n$-dimensional mod $2$ Moore spaces has a multiplicative exponent $4$ below the range of $4$ times the connectivity. As a consequence, the homotopy groups of $4n$-dimensional mod $2$ Moore spaces have an exponent $4$ below the range of $4$ times the connectivity.
Let p be an odd regular prime, and assume that the Lichtenbaum-Quillen conjecture holds for K(Z[1/p]) at p. Then the p-primary homotopy type of the smooth Whitehead spectrum Wh(*) is described. A suspended copy of the cokernel-of-J spectrum splits off, and the torsion homotopy of the remainder equals the torsion homotopy of the fiber of the restricted S^1-transfer map t: SigmaCP^infty--> S. The homotopy groups of Wh(*) are determined in a range of degrees, and the cohomology of Wh(*) is expressed as an A-module in all degrees, up to an extension. These results have geometric topological interp…
The K-theory cochains of H-spaces and height 1 chromatic homotopy theory
本文研究了H空间的K理论余弦和高度1色染 homotopy 理论，通过生成元和关系描述了KU_p余弦谱，并证明了其同伦类型的周期性建模。
Given a finite quandle, we introduce a quandle homotopy invariant of knotted surfaces in the 4-sphere, modifying that of classical links. This invariant is valued in the third homotopy group of the quandle space, and is universal among the (generalized) quandle cocycle invariants. We compute the second and third homotopy groups, with respect to "regular Alexander quandles". As a corollary, any quandle cocycle invariant using the dihedral quandle of prime order is a scalar multiple of the Mochizuki 3-cocycle invariant. As another result, we determine the third quandle homology group of the dihe…
The focus of this paper is the comparison of two unstable homotopy spectral sequences-- the unstable mod p Adams spectral sequence that computes the unstable homotopy of a p-complete space, and the Goerss--Hopkins spectral sequence, which computes the unstable homotopy of the space of E-infinity maps between Hk-algebras, where k is the algebraic closure of the field with p elements and p is an odd prime. Using an adjunction between p-complete nilpotent spaces and a subset of Hk-algebras, this paper shows that the unit of this adjunction provides an isomorphism between these spectral sequences.
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 answer a weaker version of the classification problem for the homotopy types of $(n-2)$-connected closed orientable $(2n-1)$-manifolds. Let $n\geq 6$ be an even integer, and $X$ be a $(n-2)$-connected finite orientable Poincaré $(2n-1)$-complex such that $H^{n-1}(X;\mathbb{Q})=0$ and $H^{n-1}(X;\mathbb{Z}_2)=0$. Then its loop space homotopy type is uniquely determined by the action of higher Bockstein operations on $H^{n-1}(X;\mathbb{Z}_p)$ for each odd prime $p$. A stronger result is obtained when localized at odd primes.
We study $p$-localizations, where $p$ is an odd prime, of the full subcategories $S^n$ of stable homotopy category consisting of CW-complexes having cells in $n$ successive dimensions. Using the technique of triangulated categories and matrix problems we classify atoms (indecomposable objects) in $S_p^n$ for $n\le 4(p-1)$ and show that for $n>4(p-1)$ such classification is wild in the sense of the representation theory.
Let $G$ be a simply-connected simple compact Lie group and let $M$ be an orientable smooth closed 4-manifold. In this paper we calculate the homotopy type of the suspension of $M$ and the homotopy types of the gauge groups of principal $G$-bundles over $M$ when $π_1(M)$ is: (1)~$\mathbb{Z}^{*m}$, (2)~$\mathbb{Z}/p^r\mathbb{Z}$, or (3)~$\mathbb{Z}^{*m}*(*^n_{j=1}\mathbb{Z}/p_j^{r_j}\mathbb{Z})$, where $p$ and the $p_j$'s are odd primes.
As an application of Behrens and Rezk's spectral algebra model for unstable v_n-periodic homotopy theory, we give explicit presentations for the completed E-homology of the Bousfield-Kuhn functor on odd-dimensional spheres at chromatic level 2, and compare them to the level 1 case. The latter reflects earlier work in the literature on K-theory localizations.
Given a finite and connected two-dimensional $CW$-complex $K$ with fundamental group $Π$ and second integer cohomology group $H^2(K;\mathbb{Z})$ finite of odd order, we prove that: (1) for each local integer coefficient system $α:Π\to{\rm Aut}(\mathbb{Z})$ over $K$, the corresponding twisted cohomology group $H^2(K;_α\!\mathbb{Z})$ is finite of odd order, we say order $\mathbb{C}^{\ast}(α)$, and there exists a natural function -- which resemble that one defined by the twisted degree -- from the set $[K;\mathbb{R}P^2]_α^{\ast}$ of the based homotopy classes of based maps inducing $α$ on $π_1$ i…
We give estimates for the torsion in the Postnikov sections $τ_{[1, n]} S^0$ of the sphere spectrum, and show that the $p$-localization is annihilated by $p^{n/(2p-2) + O(1)}$. This leads to explicit bounds on the exponents of the kernel and cokernel of the Hurewicz map $π_*(X) \to H_*(X; \mathbb{Z})$ for a connective spectrum $X$. Such bounds were first considered by Arlettaz, although our estimates are tighter and we prove that they are the best possible up to a constant factor. As applications, we sharpen existing bounds on the orders of $k$-invariants in a connective spectrum, sharpen boun…
Let $(G,H)=(SU(2n+1),SO(2n+1)),\,(SU(2n),Sp(n)),\,(SO(2n),SO(2n-1)),\,(E_6,F_4),\,(Spin(8),G_2)$, and let $p$ be any prime $\ge 5$ for $(G,H)=(E_6,F_4)$, any prime $p\ne 3$ for $(G,H)=(Spin(8),G_2)$, and any odd prime otherwise. The classical result of Harris on the relation between the homotopy groups of $G$ and $H$ is reinterpreted as a $p$-local homotopy equivalence $G\simeq_{(p)}H\times G/H$, which yields a projection $G_{(p)}\to H_{(p)}$. We show how much this projection preserves the higher homotopy associativity.
Odd primary analogs of Real orientations
本文在$C_p$-等变同伦论中定义了$\mu_p$-定向，推广了Real定向，并证明Morava $E$-理论和$\mathrm{tmf}(2)$分别具有$\mu_p$和$\mu_3$定向，通过单个等变映射生成其同伦，揭示高度移位现象。
We investigate the exponents of the total Cohen groups $[Ω(\mathbb S^{r+1}), Ω(Y)]$ for any $r\ge 1$. In particular, we show that for $p\ge 3$, the $p$-primary exponents of $[Ω(\mathbb S^{r+1}), Ω(\mathbb S^{2n+1})]$ and $[Ω(\mathbb S^{r+1}), Ω(\mathbb S^{2n})]$ coincide with the $p$-primary homotopy exponents of spheres $\mathbb S^{2n+1}$ and $\mathbb S^{2n}$, respectively. We further study the exponent problem when $Y$ is a space with the homotopy type of $Σ(n)/G$ for a homotopy $n$-sphere $Σ(n)$, the complex projective space $\mathbb{C}P^n$ for $n\ge 1$ or the quaternionic projective space …
In this paper we constructs a new nontrivial family in the stable homotopy groups of spheres $π_{p^nq+2pq+q-3}S$ which is of order $p$ and is represented by $k_0h_{n} \in Ext_A^{3,p^nq+2pq+q}(\mathbb{Z}_p,\mathbb{Z}_p)$ in the Adams spectral sequence, where $p\geq 5$ is an odd prime, $n\geq 3$ and $q=2(p-1)$. In the course of the proof, a new family of homotopy elements in $π_{\ast}V(1)$ which is represented by $β_{\ast}{i^{\prime}}_{\ast}i_{\ast}({h}_n)\in Ext_A^{2,p^nq+(p+1)q+1}(H^{\ast}V(1),\mathbb{Z}_p)$ in the Adams sequence is detected.
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.
Infinite families of higher torsion in the homotopy groups of Moore spaces
本文细化了 Moore 空间双环路空间的稳定 Snaith 分裂，借此在 mod $p^r$ Moore 空间同伦群中构造出阶为 $p^{r+1}$ 的无限 $v_1$-周期族，并揭示了奇素数下 Moore 谱同伦群与不稳定同伦群的直和关系。
Nilpotence in the homotopy of $\mathbb{E}_\infty$-ring spectra is detected by the classical $H\mathbb{Z}$-Hurewicz homomorphism. Inspired by questions of Mathew, Noel, and Naumann, we investigate the extent to which this criterion holds in the homotopy of $\mathbb{E}_n$-ring spectra. For all odd primes $p$ and all chromatic heights $h$, we use the Cohen-Moore-Neisendorfer theorem to construct examples of $K(h)$-local, $\mathbb{E}_{2n-1}$-algebras with non-nilpotent $p^n$-torsion. We exploit the interaction of the Bousfield-Kuhn functor on odd spheres and Rezk's logarithm to show that our bound…
We use methods of combinatorial number theory to prove that, for each $n>1$ and any prime $p$, some homotopy group $π_i(SU(n))$ contains an element of order $p^{n-1+ord_p([n/p]!)}$, where $ord_p(m)$ denotes the largest integer $α$ such that $p^α$ divides $m$.
Let E(n) and T(m) for nonnegative integers n and m denote the Johnson-Wilson and the Ravenel spectra, respectively. Given a spectrum whose E(n)_*-homology is E(n)_*(T(m))/(v_1,...,v_{n-1}), then each homotopy group of it estimates the order of each homotopy group of L_nT(m). We here study the E(n)-based Adams E_2-term of it and present that the determination of the E_2-term is unexpectedly complex for odd prime case. At the prime two, we determine the E_{infty}-term for pi_*(L_2T(1)/(v_1)), whose computation is easier than that of pi_*(L_2T(1)) as we expect.
Bousfield recently gave a formula for the odd-primary v1-periodic homotopy groups of a finite H-space in terms of its K-theory and Adams operations. We apply Bousfield's theorem to give explicit determinations of the v1-periodic homotopy groups of (E8,5) and (E8,3), thus completing the determination of all odd-primary v1-periodic homotopy groups of compact simple Lie groups, a project suggested by Mimura in 1989. The method involves no homotopy theoretic input and no spectral sequences. The input is the second eexterior power operation in the representation ring of E8, which we determine using…
We determine precisely the largest v1-periodic homotopy groups of SU(2^e) and SU(2^e + 1). This gives new results about the largest actual homotopy groups of these spaces. Our proof relies on results about 2-divisibility of restricted sums of binomial coefficients times powers proved by the author in a companion paper.
The third homotopy module of a 2-complex
该研究针对基本群为G的连通2-复形，给出了将其第三同伦群π₃(X)计算为ℤ[G]上模的方法，并证明了奇数阶有限基本群情形下π₃(X)的稳定类由基本群完全决定。
Let E be an H-space acting on a based space X. Then we refer to ev: E -> X, the map obtained by acting on the base point of X, as a ``generalized evaluation map." We establish several fundamental results about the rational homotopy behaviour of a generalized evaluation map, all of which apply to the usual evaluation map Map(X,X;1)->X. With mild hypotheses on X, we show that a generalized evaluation map ev factors, up to rational homotopy, through a map Gamma_ev: S_ev -> X where S_ev is a (relatively small) finite product of odd-dimensional spheres and the map induced by Gamma_ev on rational ho…
Let $P^{2n+1}$ be a two-cell complex which is formed by attaching a $(2n+1)$--cell to a $2m$--sphere by a suspension map. We construct a universal space $U$ for $P^{2n+1}$ in the category of homotopy associative, homotopy commutative $H$--spaces. By universal we mean that $U$ is homotopy associative, homotopy commutative, and has the property that any map $f\colon P^{2n+1}\lra Y$ to a homotopy associative, homotopy commutative $H$--space $Y$ extends to a uniquely determined $H$--map $\bar{f}\colon U\lra Y$. We then prove upper and lower bounds of the $H$--homotopy exponent of $U$. In the case …
We show that $S^n \vee S^m$ is $\mathbb{Z}/p^r$-hyperbolic for all primes $p$ and all $r \in \mathbb{Z}^+$, provided $n,m \geq 2$, and consequently that various spaces containing $S^n \vee S^m$ as a $p$-local retract are $\mathbb{Z}/p^r$-hyperbolic. We then give a $K$-theory criterion for a suspension $ΣX$ to be $p$-hyperbolic, and use it to deduce that the suspension of a complex Grassmannian $ΣGr_{k,n}$ is $p$-hyperbolic for all odd primes $p$ when $n \geq 3$ and $0<k<n$. We obtain similar results for some related spaces.
The chromatic spectral sequence is introduced in \cite{mrw} to compute the $E_2$-term of the \ANSS\ for computing the stable homotopy groups of spheres. The $E_1$-term $E_1^{s,t}(k)$ of the spectral sequence is an Ext group of $BP_*BP$-comodules. There are a sequence of Ext groups $E_1^{s,t}(n-s)$ for non-negative integers $n$ with $E_1^{s,t}(0)=E_1^{s,t}$, and Bockstein spectral sequences computing a module $E_1^{s,*}(n-s)$ from $E_1^{s-1,*}(n-s+1)$. So far, a small number of the $E_1$-terms are determined. Here, we determine the $E_1^{1,1}(n-1)=\e^1M^1_{n-1}$ for $p>2$ and $n>3$ by computing…
In this note, for each odd prime $p$, we show that the orders of the $KU$-local homotopy groups of the mod $p$ Moore spectrum are equal to denominators of special values of certain quotients of Dedekind zeta-functions of totally real number fields. With this observation in hand, we give a cute topological proof of the Leopoldt conjecture for those number fields, by showing that it is a consequence of periodicity properties of $KU$-local stable homotopy groups.
We prove analogues for odd primes of results of Snaith and Barker-Snaith. Let l denote the p-complete connective Adams summand and consider the group of left l-module automorphisms of l smash l in the stable homotopy category which induce the identity on mod p homology. We prove a group isomorphism between this group and a certain group of infinite invertible upper triangular matrices with entries in the p-adic integers. We determine information about the matrix corresponding to the automorphism 1 smash Psi^q, where Psi^q is the Adams operation and q is an integer which generates the p-adic un…
Adjacent non-stable layers in the $\mathbb{A}^1$-homotopy of the special linear tower
本研究确定特殊线性塔中相邻Stiefel纤维序列的两个跨阶段态射的性质，复实现可分类分裂二次曲面上的定向向量丛，并将结果用于给出投射模的分裂与生成准则。
Let M be a smooth manifold which is homeomorphic to the n-fold product of S^k, where k is odd. There is an induced homomorphism from the group of diffeomorphisms of M to the automorphism group of H k (M ; Z). We prove that the image of this homomorphism contains a congruence subgroup of SL_n (Z) whenever n is at least 3.
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。
Given a group $G$, its lattice of subgroups $\mathcal{L}(G)$ can be viewed as a simplicial complex in a natural way. The inclusion of $1_G, G \in \mathcal{L}(G)$ implies that $\mathcal{L}(G)$ is contractible, and so we study the topology of the order complex $\widehat{\mathcal{L}(G)} := \mathcal{L}(G) \setminus \{1_G,G\}$. In this short note we consider the homotopy type of $\widehat{\mathcal{L}(G)}$ where $G \cong PGL_2(p^{2^n})$, $p \geq 3$, $n \geq 1$ and show that $\widehat{\mathcal{L}(G)}$ is contractible. This is consistent with a conjecture of Shareshian on the homotopy type of order co…
Both the gauge groups and $5$-manifolds are important in physics and mathematics. In this paper, we combine them together to study the homotopy aspects of gauge groups over $5$-manifolds. For principal bundles over non-simply connected oriented closed $5$-manifolds of certain type, we prove various homotopy decompositions of their gauge groups according to different geometric structures on the manifolds, and give partial solution to the classification of the gauge groups. As applications, we estimate the homotopy exponents of their gauge groups, and show periodicity results of the homotopy gro…
We show that the homotopy groups of a connective $E_k$-ring spectrum with an $E_k$-cell attached along a class $α$ in degree $n$ are isomorphic to the homotopy groups of the cofiber of the self-map associated to $α$ through degree $2n$. Using this, we prove that the $2n-1^{st}$ homotopy groups of Ravenel's $X(n)$ spectra are cyclic for all $n$. This further implies that, after localizing at a prime, $X(n+1)$ is homotopically unique as the $E_1$-$X(n)$-algebra with homotopy groups in degree $2n-1$ killed by an $E_1$-cell. Lastly, we prove analogous theorems for a sequence of $E_k$-ring Thom spe…
We prove that, for p an odd prime, every finite p-group of rank 3 acts freely on a finite complex X homotopy equivalent to a product of three spheres.
On spaces of minimal higher topological complexity
本文研究满足第n阶拓扑复杂度取最小值n-1的拓扑空间分类，在去除原有幂零性条件的基础上，证明此类空间要么同伦等价于奇维球面，要么是带无限循环基本群的同调圆周。
Iterated Topological Hochschild Homology of $\mathbb{F}_p$ at odd primes
本文计算奇数素数下 $\mathbb{F}_p$ 的迭代拓扑 Hochschild 同调的同伦群，通过 bar 谱序列证明其在 $E^2$ 页坍缩，并确定环结构。
Fix an odd prime $p$ and let $X$ be the $p$-localization of a finite suspended $CW$-complex. Given certain conditions on the reduced mod-$p$ homology $\bar H_*(X;\zmodp)$ of $X$, we use a decomposition of $ΩΣX$ due to the second author and computations in modular representation theory to show there are arbitrarily large integers $i$ such that $ΩΣ^i X$ is a homotopy retract of $ΩΣX$. This implies the stable homotopy groups of $ΣX$ are in a certain sense retracts of the unstable homotopy groups, and by a result of Stanley, one can confirm the Moore conjecture for $ΣX$. Under additional assumptio…
Let $G$ be a simply-connected, compact, simple Lie group of low rank relative to a fixed prime $p$. After localization at $p$, there is a space $A$ which "generates" $G$ in a certain sense. Assuming $G$ satisfies a homotopy nilpotency condition relative to $p$, we show that the Samelson product $\langle Id_G, Id_G\rangle$ of the identity of $G$ equals the order of the Samelson product $\langle\imath,\imath\rangle$ of the inclusion $\imath:A\to G$. Applying this result, we calculate the orders of $\langle Id_G,Id_G\rangle$ for all $p$-regular Lie groups and give bounds on the orders of $\langle…
In a recent paper, it was shown that the problem of existence of a continuous map $X \to Y$ extending a given map $A \to Y$ defined on a subspace $A \subseteq X$ is undecidable, even for $Y$ an even-dimensional sphere. In the present paper, we prove that the same problem for $Y$ an odd-dimensional sphere is decidable. More generally, the same holds for any $d$-connected target space $Y$ whose homotopy groups $π_k Y$ are finite for $k>2d$.
Under certain conditions, we describe the homotopy type of the homo-topy fibre of the inclusion map F\_n(X) $\rightarrow$ $\prod$\_1^n X for the n-th configuration space F\_n(X) of a topological manifold X without boundary such that dim(X) $\ge$ 3. We then apply our results to the cases where either the universal covering of X is contractible or X is an orbit space S^k/G of a tame, free action of a Lie group G on the k-sphere S^k. If the group G is finite and k is odd, we give a full description of the long exact sequence in homotopy of the homotopy fibration of the inclusion map F\_n(S^k/G) $…
Let $p$ be prime. We prove that, for $n$ odd, the $p$-torsion part of $π_q(S^{n})$ has cardinality at most $p^{2^{\frac{1}{p-1}(q-n+3-2p)}}$, and hence has rank at most $2^{\frac{1}{p-1}(q-n+3-2p)}$. For $p=2$ these results also hold for $n$ even. The best bounds proven in the existing literature are $p^{2^{q-n+1}}$ and $2^{q-n+1}$ respectively, both due to Hans-Werner Henn. The main point of our result is therefore that the bound grows more slowly for larger primes. As a corollary of work of Henn, we obtain a similar result for the homotopy groups of a broader class of spaces.
Homotopy theoretic properties of open books
本文研究开书的同伦论性质，给出开书基点环路空间的整系数与有理分解结果，推广有理同伦论经典二歧性，并证明奇维球面Milnor开书的单值化同调诱导作用非幂零。
Given a simply connected, closed four manifold, we associate to it a simply connected, closed, spin five manifold. This leads to several consequences : the stable and unstable homotopy groups of such a four manifold is determined by its second Betti number, and the ranks of the homotopy groups can be explicitly calculated. We show that for a generic metric on such a smooth four manifold with second Betti number at least three, the number of geometrically distinct periodic geodesics of length at most l grow exponentially as a function of l. The number of closed Reeb orbits of length at most l o…
Counting homotopy classes of mappings via Dijkgraaf-Witten invariants
本文研究有限群自由作用于奇数维球面时，闭可定向流形上给定度数映射的同伦类计数问题，证明其有限性并用Dijkgraaf-Witten不变量给出基数公式。
Let $G$ be a group and $R,S,T$ its normal subgroups. There is a natural extension of the concept of commutator subgroup for the case of three subgroups $\|R,S,T\|$ as well as the natural extension of the symmetric product $\|\bf r,\bf s,\bf t\|$ for corresponding ideals $\bf r,\bf s, \bf t$ in the integral group ring $\mathbb Z[G]$. In this paper, it is shown that the generalized dimension subgroup $G\cap (1+\|\bf r,\bf s,\bf t\|)$ has exponent 2 modulo $\|R,S,T\|.$ The proof essentially uses homotopy theory. The considered generalized dimension quotient of exponent 2 is identified with a subg…
Cobordism groups of cooriented fold maps of codimension 1 are computed completely. Namely their odd torsion part coincides with that of the stable homotopy group of spheres in the same dimension, while the 2-primary part is the kernel of the Kahn-Priddy map. (The Kahn-Priddy map is an epimorhism of the stable homotopy group of the infinite dimensional real projective space onto the 2-primary part of the stable homotopy group of spheres). Analogous results - modulo small primes - are obtained for cusp maps and more complicated Morin maps as well.
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。