geometric-and-algebraic-topology

I am looking for a relatively elementary reference on geometric realization of simplicial objects, especially in connection with simplicial homotopies. The application I have in mind is to understand ...

higher geometry / derived geometry Ingredients Concepts geometric little (∞,1)-toposes geometric big (∞,1)-toposes Constructions Examples derived smooth geometry Theorems Classical groups Finite groups Group schemes Topological groups Lie groups Super-Lie groups Higher groups Cohomology and Extensions Related concepts physics, mathematical physics, philosophy of physics theory (physics), model (p…

We discuss a topological characterization of Cantor sets. Topologically speaking there is only one Cantor set. The topological characterization consists of these three properties: ….….zero-dimensional, ….….perfect, ….….compact metric space. Thus, every space satisfying this characterization is homeomorphic to the Cantor … Continue reading →

On anyon-excitations in topological superconductors. via Majorana zero modes: Original proposal: Review: Sankar Das Sarma, Michael Freedman, Chetan Nayak, Majorana Zero Modes and Topological Quantum Computation, npj Quantum Information 1, 15001 (2015) (nature:npjqi20151) Nur R. Ayukaryana, Mohammad H. Fauzi, Eddwi H. Hasdeo, The quest and hope of Majorana zero modes in topological superconductor …

Say I hand you two finite posets $P,Q$. Is there an algorithm which allows you to decide whether or not the suspensions are weak homotopy equivalent? For definiteness, you can take the suspension to ...

topology (point-set topology, point-free topology) see also differential topology, algebraic topology, functional analysis and topological homotopy theory Basic concepts fiber space, space attachment Kolmogorov space, Hausdorff space, regular space, normal space sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact Examples B…

analysis (differential/integral calculus, functional analysis, topology) metric space, normed vector space open ball, open subset, neighbourhood convergence, limit of a sequence compactness, sequential compactness … … An ordered field is real closed if it satisfies the following two properties: Any non-negative element in has a square root in ; Any odd-degree polynomial function with coefficients…

In every cohesive (∞,1)-topos there is an intrinsic notion of ∞-Chern-Weil theory that gives rise to a notion of connection on principal ∞-bundles. We describe here details of the realization of this general abstract structure in the cohesive -topos Smooth∞Grpd of smooth ∞-groupoids. For an ∞-Lie group, a connection on a smooth -principal ∞-bundle is a structure that supports the Chern-Weil homom…

An idea from topology explains why you can never get rid of your cowlicks—and, oddly enough, it’s critical in nuclear fusion

symmetric monoidal (∞,1)-category of spectra analysis (differential/integral calculus, functional analysis, topology) metric space, normed vector space open ball, open subset, neighbourhood convergence, limit of a sequence compactness, sequential compactness … … The different types of square root partial functions on the real numbers that satisfy the functional equation on some subset of the real…

Context Topology topology (point-set topology, point-free topology) see also differential topology, algebraic topology, functional analysis and topological homotopy theory Introduction Basic concepts - open subset, closed subset, neighbourhood - topological space, locale - base for the topology, neighbourhood base - finer/coarser topology - closure, interior, boundary - separation, sobriety - con…

homotopy theory, (∞,1)-category theory, homotopy type theory flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed… models: topological, simplicial, localic, … see also algebraic topology Introductions Definitions Paths and cylinders Homotopy groups Basic facts Theorems Discrete homotopy theory (also known as A-homotopy theory) is an area of mathematics concerned w…

We introduce the Montilla Knot, a closed parametric surface constructed by sweeping a periodic cross-section along the figure-eight knot (4_1) using a Bishop parallel-transport frame. The surface is defined by an explicit set of equations that guarantee closure of the cross-section and verified non-self-intersection. A key finding is that the figure-eight knot possesses an intrinsic holonomy angl…

Definitions. Category $\sf SpS$ of spectral spaces. An object in $\sf SpS$ is a spectral space, i.e. a $T_0$ compact space such that (1) any irreducible subset is the closure of a point and such that ...

I am taking an algebraic topology course, and a lot proofs we are doing right now, say of jordan curve theorem, hairy ball, brouwer fixed point theorem, tda we do using singular homology theory. ...

Synthetic Stone duality is a program for studying the synthetic topology of second countable Stone spaces and compact Hausdorff spaces, as well as a synthetic approach to the mathematics of the (infinity,1)-topos of light condensed anima from condensed mathematics.

and rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable) representation, 2-representation, ∞-representation Grothendieck group, lambda-ring, symmetric function, formal group principal bundle, torsor, vector bundle, Atiyah Lie algebroid Eilenberg-Moore category, algebra over an operad, actegory, crossed module What may be called Borel-equivarian…

On rational homotopy theory of non-nilpotent spaces via deck-Borel-equivariant rational homotopy theory of their universal covering spaces: Antonio Gómez-Tato, Stephen Halperin, Daniel Tanré: Rational homotopy theory for non-simply connected spaces, Trans. Amer. Math. Soc. 352 (2000) 1493–1525 [jstor:118074, doi:10.1090/S0002-9947-99-02463-0] Urtzi Buijs, Yves Félix, Aniceto Murillo, Daniel Tanré…