Alphabeta MathSpectra and Stable Homotopy Groups
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✓ 6 results · all verified · 6 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Algebraic Topology
Linear Algebra
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- Topology of ℝ
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Filters and Ultrafilters
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Definition
Compactly generated based spaces and well-pointed objects
Definition
Smash product of based spaces
Proposition
Canonical associativity, symmetry, and unit maps for smash products
Definition
Sequential prespectra, spectra, and adjoint structure maps
Definition
Suspension and sphere prespectra
Definition
Strict maps and structure-compatible homotopies of sequential prespectra
Definition
Stable homotopy groups of a sequential prespectrum
Lemma
Stable homotopy colimits are independent of a cofinal tail
Proposition
Strict prespectrum maps act functorially on stable homotopy groups
Definition
Shift and suspension of sequential prespectra
Definition
Stable stems of the sphere
Lemma
Freudenthal identifies the eventual suspension system for spheres
Proposition
The sphere prespectrum groups are the classical stable stems
Definition
Pairings and unital multiplication of sequential prespectra
Proposition