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Stable homotopy groups of spheres: from dimension 0 to 90
Using techniques in motivic homotopy theory, especially the theorem of Gheorghe, the second and the third author on the isomorphism between motivic...
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Homotopy type through homology groups
We show that for certain pairs of nonnegative integers, if the reduced integral homology groups of a space are nonzero only in that pair of degrees...
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Four Families of Nontrivial Product Elements in the Stable Homotopy Groups of Spheres
In this paper, the authors introduce a new effective method to compute the generators of the E 1 -term of the May spectral sequence. This helps them to...
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2-local unstable homotopy groups of indecomposable A
3 2 -complexesIn this paper, we calculate the 2-local unstable homotopy groups of indecomposable A
3 2 -complexes. The main technique used is analysing the homotopy... -
Homotopy Theory: Fundamental Group and Higher Homotopy Groups
This chapter officially inaugurates homotopy theory to begin a study of algebraic topology by conveying the basic concepts of homotopy and...
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Homotopy Spheres
This is a beautiful and illuminating example. It shows how the general surgery methods, which we have developed so far, apply to a concrete problem,...
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Homotopy, homology, and persistent homology using closure spaces
We develop persistent homology in the setting of filtrations of (Čech) closure spaces. Examples of filtrations of closure spaces include metric...
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Nonexistence of Colimits in Naive Discrete Homotopy Theory
We show that the quasicategory defined as the localization of the category of (simple) graphs at the class of A-homotopy equivalences does not admit...
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The cube axiom and resolutions in homotopy theory
We show that a version of the cube axiom holds in cosimplicial unstable coalgebras and cosimplicial spaces equipped with a resolution model...
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Homotopy Sheaves on Generalised Spaces
We study the homotopy right Kan extension of homotopy sheaves on a category to its free cocompletion, i.e. to its category of presheaves. Any...
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From samples to persistent stratified homotopy types
The natural occurrence of singular spaces in applications has led to recent investigations on performing topological data analysis (TDA) in a...
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Smashing localizations in equivariant stable homotopy
We study how smashing Bousfield localizations behave under various equivariant functors. We show that the analogs of the smash product and chromatic...
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On Homotopy Classification of Elliptic Problems with Contractions and K-Groups of Corresponding C*-Algebras
We consider a computation of a group of stable homotopy classes for pseudodifferential elliptic boundary problems. We study this problem in terms of...
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Homotopy cardinality via extrapolation of Morava–Euler characteristics
We answer a question of John Baez, on the relationship between the classical Euler characteristic and the Baez–Dolan homotopy cardinality, by...
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K-flatness and orthogonality in homotopy categories
K-flatness for unbounded complexes of modules over a ring R was introduced by Spaltenstein [27], as an analogue of the classical notion of flatness...
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Homotopy
There’s an intuitive notion of ‘equivalence of shapes’ in topology that is broader than the concept of homeomorphism. Two connected regular subsets...
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A Whitehead theorem for periodic homotopy groups
We show that v n -periodic homotopy groups detect homotopy equivalences between simply-connected finite CW-complexes.
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Surjectivity of the Étale Excision Map for Homotopy Invariant Framed Presheaves
AbstractThe categories of framed correspondences, framed presheaves, and framed sheaves were introduced by V. Voevodsky in his foundational “Notes...