Correction: Stable homotopy classification of $BG_{p}^{\wedge}$

Non-factorisation of Arf-Kervaire classes through ${\mathbb RP}^{\infty} \wedge {\mathbb RP}^{\infty}$

As an application of the upper triangular technology method of (V.P. Snaith: {\em Stable homotopy -- around the Arf-Kervaire invariant}; Birkhäuser Progress on Math. Series vol. 273 (April 2009)) it is shown that there do not exist stable homotopy classes of $ {\mathbb RP}^{\infty} \wedge {\mathbb RP}^{\infty}$ in dimension $2^{s+1}-2$ with $s \geq 2$ whose composition with the Hopf map to $ {\mathbb RP}^{\infty}$ followed by the Kahn-Priddy map gives an element in the stable homotopy of spheres of Arf-Kervaire invariant one.

The $v_n$-periodic Goodwillie tower on Wedges and Cofibres

We introduce general methods to analyse the Goodwillie tower of the identity functor on a wedge $X \vee Y$ of spaces (using the Hilton-Milnor theorem) and on the cofibre $\mathrm{cof}(f)$ of a map $f: X \rightarrow Y$. We deduce some consequences for $v_n$-periodic homotopy groups: whereas the Goodwillie tower is finite and converges in periodic homotopy when evaluated on spheres (Arone-Mahowald), we show that neither of these statements remains true for wedges and Moore spaces.

缩小的球面楔形的Čech同调群

The Čech homotopy groups of a shrinking wedge of spheres

本文研究了缩小球面楔形的Čech同调群，推导了其同构关系，并探讨了自然同态的像结构。

The same $n$-type structure of the suspension of the wedge products of the Eilenberg-MacLane spaces

For a connected CW-complex, we let $SNT(X)$ be the set of all homotopy types $[Y]$ such that the Postnikov approximations $X^{(n)}$ and $Y^{(n)}$ of $X$ and $Y$, respectively, are homotopy equivalent for all positive integers $n$. In 1992, McGibbon and Møller (\cite[page 287]{MM}) raised the following question: Is $SNT(Σ\mathbb C P^\infty) = *$ or not? In this article, we give an answer to the more generalized version of this query: The set of all the same $n$-types of the suspended wedge sum of the Eilenberg-MacLane spaces of various types of both even and odd integers is the set which consis…

关于小有限空间的弱同伦类型

On the weak homotopy types of small finite spaces

本文证明了具有12个或更少点的连通有限拓扑空间（或连通有限偏序集的阶复形）具有球束的弱同伦类型。

The homotopy type of skeleta of the flag complex over a finite vector space

The aim of this paper is to give a (discrete) Morse theoretic proof of the fact that the $k$-th skeleton of the flag complex $\mathcal{F}$, associated to the lattice of subspaces of a finite dimensional vector space, is homotopy equivalent to a wedge of spheres of dimension $\min\{k,\dim(\mathcal{F})\}$. The tight control provided by Morse theoretic methods allows us to give an explicit formula for the number of spheres appearing in each of these wedge summands.

Homotopy Type of Independence Complexes of Certain Families of Graphs

We show that the independence complexes of generalised Mycielskian of complete graphs are homotopy equivalent to a wedge sum of spheres, and determine the number of copies and the dimensions of these spheres. We also prove that the independence complexes of categorical product of complete graphs are wedge sum of circles, upto homotopy. Further, we show that if we perturb a graph $G$ in a certain way, then the independence complex of this new graph is homotopy equivalent to the suspension of the independence complex of $G$.

On the Homotopy Type of the Polyhedral Join over the Independence Complex of a Forest

We consider a certain class of simplicial complexes which includes the independence complexes of forests. We show that if a simplicial complex $K$ belongs to this class, then the polyhedral join $\mathcal{Z}^*_{K}(\underline{X}, \emptyset)$ is homotopy equivalent to a wedge sum of CW complexes of the form $Σ^r X_{i_1} * X_{i_2} * \cdots * X_{i_k}$, where $\underline{X}$ is a family $\{X_i\}_{i \in V(K)}$ of CW complexes and $Σ$ denotes the unreduced suspension. This result is applied to study the homotopy type of the independence complex of the lexicographic product $G[H]$ of a graph $H$ over …

Homotopy types of moment-angle complexes for flag complexes

We study the homotopy types of moment-angle complexes, or equivalently, of complements of coordinate subspace arrangements. The overall aim is to identify the simplicial complexes K for which the corresponding moment-angle complex Z_K has the homotopy type of a wedge of spheres or a connected sum of sphere products. When K is flag, we identify in algebraic and combinatorial terms those K for which Z_K is homotopy equivalent to a wedge of spheres, and give a combinatorial formula for the number of spheres in the wedge. This extends results of Berglund and Joellenbeck on Golod rings and homotopy…

Every K(n)-local spectrum is the homotopy fixed points of its Morava module

Let n \geq 1 and let p be any prime. Also, let E_n be the Lubin-Tate spectrum, G_n the extended Morava stabilizer group, and K(n) the nth Morava K-theory spectrum. Then work of Devinatz and Hopkins and some results due to Behrens and the first author of this note, show that if X is a finite spectrum, then the localization L_{K(n)}(X) is equivalent to the homotopy fixed point spectrum (L_{K(n)}(E_n \wedge X))^{hG_n}, which is formed with respect to the continuous action of G_n on L_{K(n)}(E_n \wedge X). In this note, we show that this equivalence holds for any (S-cofibrant) spectrum X. Also, we…

$\mathbb{R}P^2$ 和 $\mathbb{R}P^2 \wedge \mathbb{C}P^2$ 的 $\operatorname{E}_2^{hC_6}$-同调

The $\operatorname{E}_2^{hC_6}$-homology of $\mathbb{R}P^2$ and $\mathbb{R}P^2 \wedge \mathbb{C}P^2$

本文利用同伦不动点谱序列，计算了素数2处高度2的Morava E-theory下 $\operatorname{E}_2^{hC_6}$ 与 $\mathbb{R}P^2$ 及 $\mathbb{R}P^2 \wedge \mathbb{C}P^2$ 碎积的同伦群。

The homotopy classification of four-dimensional toric orbifolds

Let $X$ be a $4$-dimensional toric orbifold. If $H^3(X)$ has a non-trivial odd primary torsion, then we show that $X$ is homotopy equivalent to the wedge of a Moore space and a CW-complex. As a corollary, given two 4-dimensional toric orbifolds having no 2-torsion in the cohomology, we prove that they have the same homotopy type if and only their integral cohomology rings are isomorphic.

On bipartite circle graphs and Khovanov homology

We prove that independence complex of a bipartite circle graph is homotopy equivalent to a wedge of spheres, resolving a conjecture posed by Przytycki and Silvero. As a corollary, we obtain that extreme Khovanov spectrum, $\mathcal{X}_{j_{extreme}}$ is homotopy equivalent to a wedge of spheres. In particular, the extreme Khovanov homology has no torsion.

A construction of some objects in many base cases of an Ausoni-Rognes conjecture

Let $p$ be a prime, $n \geq 1$, $K(n)$ the $n$th Morava $K$-theory spectrum, $\mathbb{G}_n$ the extended Morava stabilizer group, and $K(A)$ the algebraic $K$-theory spectrum of a commutative $S$-algebra $A$. For a type $n+1$ complex $V_n$, Ausoni and Rognes conjectured that (a) the unit map $i_n: L_{K(n)}(S^0) \to E_n$ from the $K(n)$-local sphere to the Lubin-Tate spectrum induces a map \[K(L_{K(n)}(S^0)) \wedge v_{n+1}^{-1}V_n \to (K(E_n))^{h\mathbb{G}_n} \wedge v_{n+1}^{-1}V_n\] that is a weak equivalence, where (b) since $\mathbb{G}_n$ is profinite, $(K(E_n))^{h\mathbb{G}_n}$ denotes a co…

The Simplicial EHP Sequence in A1-Algebraic Topology

We give a tool for understanding simplicial desuspension in A1-algebraic topology: we show that $X \to Ω(S^1 \wedge X)\to Ω(S^1 \wedge X \wedge X)$ is a fiber sequence up to homotopy in 2-localized A1-algebraic topology for $X = (S^1)^m \wedge G_m^{\wedge q}$ with $m>1$. It follows that there is an EHP sequence spectral sequence $$Z_{(2)} \otimes π_{n+1+i}^{A1} (S^{2n+2m+1} \wedge (G_m)^{\wedge 2q}) \Rightarrow Z_{(2)} \otimes π_{i}^{A1, s} (S^{m} \wedge (G_m)^{\wedge q}).$$

Two-complete stable motivic stems over finite fields

Let $\ell$ be a prime and $q = p^ν$ where $p$ is a prime different from $\ell$. We show that the $\ell$-completion of the $n$th stable homotopy group of spheres is a summand of the $\ell$-completion of the $(n, 0)$ motivic stable homotopy group of spheres over the finite field with $q$ elements $F_q$. With this, and assisted by computer calculations, we are able to explicitly compute the two-complete stable motivic stems $π_{n, 0}(F_q)^{\wedge}_2$ for $0\leq n\leq 18$. Additionally, we compute $π_{19, 0}(F_q)^{\wedge}_2$ and $π_{20, 0}(F_q)^{\wedge}_2$ when $q \equiv 1 \bmod 4$ assuming Morel'…

同调型的等效射影空间和格拉斯曼流形的有理稳定性类型

Rational stable homotopy type of equivariant projective spaces and Grassmannians

本文研究了等效复射影空间和格拉斯曼流形在复表示下的有理稳定性类型，通过显式分裂展示了其结构特征。

图的某些范畴积的独立复形的同伦型

Homotopy type of the independence complex of some categorical products of graphs

本文计算若干图范畴积独立复形的同伦型，证明了 $K_2\ imes K_3\ imes K_n$ 对应球面楔和猜想，并刻画了 $C_m\ imes K_n$ 的相关结构。

Higher Whitehead products in toric topology

In this paper we study the relationship between the moment-angle complex Z_k and the Davis-Januskiewicz space DJ(K) for a class of complexes K named missing-face complexes. If K has n vertices we consider the homotopy fibration sequence Z_k --> DJ(K) --> M where M is a product of n copies of infinite complex projective space. We observe that for such K, Z_k is homotopy equivalent to a wedge of spheres, and then show that under this equivalence the map Z_k --> DJ(K) is homotopic to a wedge sum of higher Whitehead products and iterated Whitehead products.

On genera of polyhedra

We consider genera of polyhedra (finite cell complexes) in the stable homotopy category. Namely, the genus of a polyhedron X is the class of polyhedra Y such that all localizations of Y are stably isomorphic to the corresponding localizations of X. We prove that Y is in the genus of X if and only if the wedge XvB is stably isomorphic to YvB, where B is the wedge of all spheres S^n such that the n-th stable homotopy group of X is not torsion. We also prove that if XvX and XvY are stably isomorphic, so are also X and Y. Several examples of calculations of genera are considered.

ψ^3 as an upper triangular matrix

In the 2-local stable homotopy category the group of left-bu-module automorphisms of bu\wedge bo which induce the identity on mod 2 homology is isomorphic to the group of infinite upper triangular matrices with entries in the 2-adic integers. We identify the conjugacy class of the matrix corresponding to 1\wedgeψ^3, where ψ^3 is the Adams operation.

Nerve complexes of circular arcs

We show that the nerve complex of n arcs in the circle is homotopy equivalent to either a point, an odd-dimensional sphere, or a wedge sum of spheres of the same even dimension. Moreover this homotopy type can be computed in time O(n log n). For the particular case of the nerve complex of evenly-spaced arcs of the same length, we determine the dihedral group action on homology, and we relate the complex to a cyclic polytope with n vertices. We give three applications of our knowledge of the homotopy types of nerve complexes of circular arcs. First, we use the connection to cyclic polytopes to …

Corrections to "Homotopy theory of nonsymmetric operads, I, II"

We correct a mistake in the construction of push-outs along free morphisms of algebras over a nonsymmetric operad in arXiv:1101.1634 [math.AT], and we fix the affected results in arXiv:1101.1634 [math.AT] and arXiv:1304.6641 [math.AT].

The equivariant stable homotopy theory around isometric linear maps

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…

An sl_n stable homotopy type for matched diagrams

There exists a simplified Bar-Natan Khovanov complex for open 2-braids. The Khovanov cohomology of a knot diagram made by gluing tangles of this type is therefore often amenable to calculation. We lift this idea to the level of the Lipshitz-Sarkar stable homotopy type and use it to make new computations. Similarly, there exists a simplified Khovanov-Rozansky sl_n complex for open 2-braids with oppositely oriented strands and an even number of crossings. Diagrams made by gluing tangles of this type are called matched diagrams, and knots admitting matched diagrams are called bipartite knots. To …

Freeness and equivariant stable homotopy

We introduce a notion of freeness for $RO$-graded equivariant generalized homology theories, considering spaces or spectra $E$ such that the $R$-homology of $E$ splits as a wedge of the $R$-homology of induced virtual representation spheres. The full subcategory of these spectra is closed under all of the basic equivariant operations, and this greatly simplifies computation. Many examples of spectra and homology theories are included along the way. We refine this to a collection of spectra analogous to the pure and isotropic spectra considered by Hill--Hopkins--Ravenel. For these spectra, the …

3-切割复形的幂环图的壳性

Shellability of 3-cut complexes of powers of cycle graphs

研究3-切割复形Δ_3(C_n^p)的壳性，证明其同调等价于多个球面的 wedge，提供显式壳序并计数生成元数量。

Corrigendum to "Groupoids, the Phragmen-Brouwer Property, and the Jordan Curve Theorem", J. Homotopy and Related Structures 1 (2006) 175-183

Omar Antol{\'ı}n Camarena pointed out a gap in the proofs in \cite{Brown15,BrownJordan} of a condition for the Phragmen--Brouwer Property not to hold; this note gives the correction in terms of a result on a pushout of groupoids, and some additional background.

Discretized configurations and partial partitions

We show that the discretized configuration space of $k$ points in the $n$-simplex is homotopy equivalent to a wedge of spheres of dimension $n-k+1$. This space is homeomorphic to the order complex of the poset of ordered partial partitions of $\{1,...,n+1\}$ with exactly $k$ parts. We compute the exponential generating function for the Euler characteristic of this space in two different ways, thereby obtaining a topological proof of a combinatorial recurrence satisfied by the Stirling numbers of the second kind.