{"title":"怀特海积和无穷和的同一性","authors":"Jeremy Brazas","doi":"arxiv-2408.10430","DOIUrl":null,"url":null,"abstract":"Whitehead products and natural infinite sums are prominent in the higher\nhomotopy groups of the $n$-dimensional infinite earring space $\\mathbb{E}_n$\nand other locally complicated Peano continua. In this paper, we derive general\nidentities for how these operations interact with each other. As an\napplication, we consider a shrinking-wedge $X$ of $(n-1)$-connected finite\nCW-complexes $X_1,X_2,X_3,\\dots$ and compute the infinite-sum closure\n$\\mathcal{W}_{2n-1}(X)$ of the set of Whitehead products $[\\alpha,\\beta]$ in\n$\\pi_{2n-1}\\left(X\\right)$ where $\\alpha,\\beta\\in\\pi_n(X)$ are represented in\nrespective sub-wedges that meet only at the basepoint. In particular, we show\nthat $\\mathcal{W}_{2n-1}(X)$ is canonically isomorphic to\n$\\prod_{j=1}^{\\infty}\\left(\\pi_{n}(X_j)\\otimes \\prod_{k>j}\\pi_n(X_k)\\right)$.\nThe insight provided by this computation motivates a conjecture about the\nisomorphism type of the elusive groups $\\pi_{2n-1}(\\mathbb{E}_n)$, $n\\geq 2$.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"26 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Identities for Whitehead products and infinite sums\",\"authors\":\"Jeremy Brazas\",\"doi\":\"arxiv-2408.10430\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Whitehead products and natural infinite sums are prominent in the higher\\nhomotopy groups of the $n$-dimensional infinite earring space $\\\\mathbb{E}_n$\\nand other locally complicated Peano continua. In this paper, we derive general\\nidentities for how these operations interact with each other. As an\\napplication, we consider a shrinking-wedge $X$ of $(n-1)$-connected finite\\nCW-complexes $X_1,X_2,X_3,\\\\dots$ and compute the infinite-sum closure\\n$\\\\mathcal{W}_{2n-1}(X)$ of the set of Whitehead products $[\\\\alpha,\\\\beta]$ in\\n$\\\\pi_{2n-1}\\\\left(X\\\\right)$ where $\\\\alpha,\\\\beta\\\\in\\\\pi_n(X)$ are represented in\\nrespective sub-wedges that meet only at the basepoint. In particular, we show\\nthat $\\\\mathcal{W}_{2n-1}(X)$ is canonically isomorphic to\\n$\\\\prod_{j=1}^{\\\\infty}\\\\left(\\\\pi_{n}(X_j)\\\\otimes \\\\prod_{k>j}\\\\pi_n(X_k)\\\\right)$.\\nThe insight provided by this computation motivates a conjecture about the\\nisomorphism type of the elusive groups $\\\\pi_{2n-1}(\\\\mathbb{E}_n)$, $n\\\\geq 2$.\",\"PeriodicalId\":501119,\"journal\":{\"name\":\"arXiv - MATH - Algebraic Topology\",\"volume\":\"26 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Algebraic Topology\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.10430\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Algebraic Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.10430","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Identities for Whitehead products and infinite sums
Whitehead products and natural infinite sums are prominent in the higher
homotopy groups of the $n$-dimensional infinite earring space $\mathbb{E}_n$
and other locally complicated Peano continua. In this paper, we derive general
identities for how these operations interact with each other. As an
application, we consider a shrinking-wedge $X$ of $(n-1)$-connected finite
CW-complexes $X_1,X_2,X_3,\dots$ and compute the infinite-sum closure
$\mathcal{W}_{2n-1}(X)$ of the set of Whitehead products $[\alpha,\beta]$ in
$\pi_{2n-1}\left(X\right)$ where $\alpha,\beta\in\pi_n(X)$ are represented in
respective sub-wedges that meet only at the basepoint. In particular, we show
that $\mathcal{W}_{2n-1}(X)$ is canonically isomorphic to
$\prod_{j=1}^{\infty}\left(\pi_{n}(X_j)\otimes \prod_{k>j}\pi_n(X_k)\right)$.
The insight provided by this computation motivates a conjecture about the
isomorphism type of the elusive groups $\pi_{2n-1}(\mathbb{E}_n)$, $n\geq 2$.