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 $\mathbb{H}P^n$ for $1\le n\le \infty$.
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.
We propose a new method to compute the $C_{2^n}$-equivariant homotopy groups of the Eilenberg-Mac Lane spectrum $H\underline{\mathbb{Z}}$ as a $RO(C_{2^n})$-graded Green functor using the generalized Tate squares. As an example, we completely compute the $C_4$ case and investigate two $\mathscr{P}$-homotopy limit spectral sequences for the family $\mathscr{P}=\{e,C_2\}$.
Ganea proved that the loop space of $\mathbb{C}P^n$ is homotopy commutative if and only if $n=3$. We generalize this result to that the loop spaces of all irreducible Hermitian symmetric spaces but $\mathbb{C}P^3$ are not homotopy commutative. The computation also applies to determining the homotopy nilpotency of the loop spaces of flag manifolds.
We construct a space $\mathbb{P}$ for which the canonical homomorphism $π_1(\mathbb{P},p) \rightarrow \checkπ_1(\mathbb{P},p)$ from the fundamental group to the first Čech homotopy group is not injective, although it has all of the following properties: (1) $\mathbb{P}\setminus\{p\}$ is a 2-manifold with connected non-compact boundary; (2) $\mathbb{P}$ is connected and locally path connected; (3) $\mathbb{P}$ is strongly homotopically Hausdorff; (4) $\mathbb{P}$ is homotopically path Hausdorff; (5) $\mathbb{P}$ is 1-UV$_0$; (6) $\mathbb{P}$ admits a simply connected generalized covering space …
We calculate $π_*K(\mathbb S)[1/2]$, the homotopy groups of $K(\mathbb S)$ away from 2, in terms of the homotopy groups of $K(\mathbb Z)$, the homotopy groups of ${\mathbb C}P^\infty_{-1}$, and the homotopy groups of $\mathbb S$. This builds on the work of Waldhausen, who computed the rational homotopy groups (building on work of Quillen and Borel) and Rognes, who calculated the groups at regular primes in terms of the homotopy groups of ${\mathbb C}P^\infty_{-1}$, and the homotopy groups of $\mathbb S$.
Let $G$ be a compact connected Lie group and $n\geqslant 1$ an integer. Consider the space of ordered commuting $n$-tuples in $G$, $Hom(\mathbb{Z}^n,G)$, and its quotient under the adjoint action, $Rep(\mathbb{Z}^n,G):=Hom(\mathbb{Z}^n,G)/G$. In this article we study and in many cases compute the homotopy groups $π_2(Hom(\mathbb{Z}^n,G))$. For $G$ simply--connected and simple we show that $π_2(Hom(\mathbb{Z}^2,G))\cong \mathbb{Z}$ and $π_2(Rep(\mathbb{Z}^2,G))\cong \mathbb{Z}$, and that on these groups the quotient map $Hom(\mathbb{Z}^2,G)\to Rep(\mathbb{Z}^2,G)$ induces multiplication by the …
We show that all homotopy $\mathbb{C}P^n$s, smooth closed manifolds with the oriented homotopy type of $\mathbb{C}P^n$, admit almost complex structures for $3 \leq n \leq 6$, and classify these structures by their Chern classes. Our methods provide a new proof of a result of Libgober and Wood on the classification of almost complex structures on homotopy $\mathbb{C}P^4$s.
We study the mod-$\ell$ homotopy type of classifying spaces for commutativity, $B(\mathbb{Z}, G)$, at a prime $\ell$. We show that the mod-$\ell$ homology of $B(\mathbb{Z}, G)$ depends on the mod-$\ell$ homotopy type of $BG$ when $G$ is a compact connected Lie group, in the sense that a mod-$\ell$ homology isomorphism $BG \to BH$ for such groups induces a mod-$\ell$ homology isomorphism $B(\mathbb{Z}, G) \to B(\mathbb{Z}, H)$. In order to prove this result, we study a presentation of $B(\mathbb{Z}, G)$ as a homotopy colimit over a topological poset of closed abelian subgroups, expanding on an …
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.
We approach a problem of realising algebraic objects in a certain universal equivariant stable homotopy theory; the global homotopy theory of Schwede. Specifically, for a global ring spectrum $R$, we consider which classes of ring homomorphisms $η_\ast\colonπ_\ast^e R\rightarrow S_\ast$ can be realised by a map $η\colon R\rightarrow S$ in the category of global $R$-modules, and what multiplicative structures can be placed on $S$. If $η_\ast$ witnesses $S_\ast$ as a projective $π_\ast^e R$-module, then such an $η$ exists as a map between homotopy commutative global $R$-algebras. If $η_\ast$ is …
In this paper, we study the homotopy groups of a shrinking wedge $X$ of a sequence $\{X_j\}$ of non-simply connected CW-complexes. Using a combination of generalized covering space theory and shape theory, we construct a canonical homomorphism $$Θ:π_n(X)\to\prod_{j\in\mathbb{N}}\bigoplus_{π_1(X)/π_1(X_j)}π_n(X_j),$$ characterize its image, and prove that $Θ$ is injective whenever each universal cover $\widetilde{X}_j$ is $(n-1)$-connected. These results (1) provide a characterization of the $n$-th homotopy group of the shrinking wedge of copies of $\mathbb{RP}^n$, (2) provide a characterizatio…
