topology

In my procrastination, I've been trying to understand some basic topology which started with the definition of topological spaces, which I pretty much understand with the definition via open sets on ...

The homology of (iterated) based loop spaces (ordinary homology or generalized homology) carries special structure, reflecting the ∞-group-structure of based loop spaces. In particular, under mild technical conditions (see Milnor-Moore 65, p. 262, Halperin 92) the Pontrjagin ring-structure induced by concatenation of loops enhances the homology coalgebra induced by the diagonal maps to that of a …

J. Peter May is a homotopy theorist at the University of Chicago, inventor of operads as a technique for studying infinite loop spaces and spectra. Peter May’s work makes extensive use of enriched- and model-category theory as power tools in algebraic topology/homotopy theory, notably in discussion of highly structured spectra in MMSS00‘s Model categories of diagram spectra (for exposition see In…

_Zenodo_. 2026Essay V of the Gradient Fractals suite executes the Topological layer of the ten-layer derivational chain. The four preceding essays established the Gradient Fractal Field’s ontological necessity (GF-I), algebraic-computational spine (GF-II), geometric character D = 93/40 (GF-III), and informational constitution dS/dτ = log₂(3) (GF-IV). GF Essay V now asks: what is the topological c…

This was asked before, but no answer was given. I would like to give a name to a manifold that can be covered by a single coordinate chart. Something other than "coordinate chart". In the ...

A connective spectrum is a connective object in the stable -category of spectra, hence a spectrum whose homotopy groups in all negative degrees are trivial: . These are equivalently: Connective spectra form a sub-(∞,1)-category of spectra There are objects in Spectra, though, that do not come from “naively” delooping a topological space infinitely many times. These are the non-connective spectra.…

Full Project Portfolio: https://drive.google.com/drive/folders/1fdKdo3edGqXVx95IntIumXlzKq22s-yw?usp=drive_link

Developmental stability is a sensitive indicator of environmental quality, and fluctuating asymmetry of bilateral morphological traits offers a straightforward means of its assessment. In plants, leaf blades are convenient structures for such analyses, yet traditional manual morphometry is laborious, subjective, and prone to interoperator variability. This study presents a fully automated compute…

vector bundle, 2-vector bundle, (∞,1)-vector bundle real, complex/holomorphic, quaternionic The basic line bundle on the 2-sphere is the complex line bundle on the 2-sphere whose first Chern class is a generator , equivalently the tautological line bundle on the Riemann sphere regarded as complex projective 1-space. This is the pullback bundle of the map to the classifying space/Eilenberg-MacLane…

This paper studies a local and explicit question: after the minimal noncom mutative neck algebra, if algebraic structure is allowed to be released through finitely many condensation layers, which three-step condensation paths can si multaneously satisfy representation capacity, center stability, zero redundancy, low cross-coupling, correct terminal gauge-Lie hierarchy, and absence of simple group…

Furuta’s theorem (also called 10/8 theorem) describes which indefinite even intersection forms arise from oriented closed smooth 4-manifolds. (According to Freedman's classification, all can from simply connected oriented closed topological 4-manifolds, which makes smoothness the crucial property.) It comes down to the forms: according to Serre’s classification theorem with Rokhlin's theorem requ…

On the action of the modular group on spin structures over closed surfaces in relation to theta functions and string amplitudes: On the canonical/geometric quantization of D=3 Chern-Simons theory: Early argument that the RR-field flux density-expressions for D-brane charge are of the form of Chern characters on topological K-theory, leading to the K-theory classification of D-brane charge: Michae…

This study defines an extended Neutrosophic b-Metric Space (NbMS) and illustrates the characteristics that are essential to its structure. The Fixed Point (FP) theorem has therefore been proved in the context of these Extended Neutrosophic b-Metric Spaces (ENbMS). Our results show symmetrical patterns and features within these mathematical frameworks, both extending and generalizing the results f…

We study the structure of solutions to the exponential Diophantine equation$$a^n+b^n=n^c+n^d,\qquad a,b,c,d,n\in\N$$under the condition that all variables are prime numbers. Through elementary number-theoretic analysis and modular constraints, we prove that if $a,b,c,d,n$ are all prime and satisfy the above equation, then necessarily $a=b=c=d=n$. This result shows that the prime constraint reduce…

Special and general types group cohomology, nonabelian group cohomology, Lie group cohomology cohomology with constant coefficients / with a local system of coefficients Special notions Variants differential cohomology Extra structure Operations Theorems representation, 2-representation, ∞-representation Grothendieck group, lambda-ring, symmetric function, formal group principal bundle, torsor, v…

By replacing the most fundamental concept in topology, Peter Scholze and Dustin Clausen are taking the first step in a far bigger program to understand why numbers behave the way they do. The post Two Researchers Are Rebuilding Mathematics From the Ground Up first appeared on Quanta Magazine

On delooping in homotopy type theory and application to Eilenberg-MacLane space types and Steenrod operations: On projective spaces in synthetic algebraic geometry: On the universal fibration of (infinity,1)-categories: On synthetic algebraic geometry:

Abstract: We present a systematic study of the Numerical Inversion Operation: given a positive integer n, we form its digit-reversal n′ by writing the digits of nin reverse order, and dene the inversion dierence Δ(n) = |n′ − n|. We prove that Δ(n) is always divisible by 9, with the exact multiple determined solely bythe outermost digits of n. For a two-digit number with tens digit d0 and units di…