| homology (mathematics) | |
|---|---|
| Name | Homology |
| Field | Algebraic topology |
| Introduced | 20th century |
| Related | Cohomology, Homotopy theory |
homology (mathematics)
Homology (mathematics) is an algebraic formalism that assigns abelian groups or modules to topological spaces, measuring their global topology via cycles and boundaries. In the context of quantum physics and especially Topological quantum field theory (TQFT), homology provides invariants used to classify phase structure, compute quantum invariants, and construct state spaces for quantum systems.
Homology arose within algebraic topology through work by Henri Poincaré, Emmy Noether, and others who formalized the relation between geometry and algebra. In quantum physics, homological tools appear in Topological quantum computing proposals (e.g., Kitaev model, Toric code), in the algebraic description of Anyons and braid group representations studied at Microsoft Research and academic groups, and in the axiomatic formulation of Topological quantum field theory by Michael Atiyah and Graeme Segal. Homology groups underpin constructions such as Floer homology, which links to quantum field theory via gauge-theoretic and string-theoretic dualities studied at institutions like Princeton University and Institut des Hautes Études Scientifiques.
At base, homology uses chain complexes of free abelian groups with boundary operators to produce homology groups H_n(X) = Z_n(X)/B_n(X). The theory is formalized in the language of category theory and homological algebra as developed by Samuel Eilenberg and Saunders Mac Lane. Key algebraic constructs include chain maps, exact sequences, and the Universal Coefficient Theorem and Künneth theorem, which relate homology to tensor products and Ext functors. In quantum applications, homology groups often serve as graded vector spaces that model spaces of quantum states, with graded dimensions related to quantum invariants like the Jones polynomial via categorification frameworks such as Khovanov homology introduced by Mikhail Khovanov.
Several homology theories provide computational and conceptual flexibility. Singular homology applies to arbitrary topological spaces and is foundational in theoretical work at universities like University of Cambridge. Simplicial homology is combinatorial and used in discrete models, relevant to lattice realizations of quantum systems such as lattice gauge theory studied at CERN and Perimeter Institute. Cellular homology exploits CW-complex structures and is efficient for spaces constructed in TQFT axioms. These variants are linked by theorems guaranteeing equivalence on well-behaved spaces, and they interact with other invariants like Euler characteristic and Poincaré duality, which are central in the classification of topological phases researched by groups at Harvard University and MIT.
In TQFT, homology enters both as an algebraic input and as a target for quantum invariants. The axiomatic TQFT framework of Michael Atiyah formalizes how cobordism categories map to vector spaces; homology and cohomology theories provide examples of such mappings. Homological knot invariants—most prominently Khovanov homology—categorify polynomial invariants like the Jones polynomial and connect to Chern–Simons theory studied in mathematical physics at Caltech and Institute for Advanced Study. Heegaard Floer homology and Seiberg–Witten Floer homology are used to extract quantum-topological information about 3- and 4-manifolds, informing research on quantum gravity models pursued at Perimeter Institute and in string theory groups at CERN and SLAC National Accelerator Laboratory.
Computational homology uses algorithms and software to compute homology groups for large complexes. Tools like CHomP, Dionysus, and Perseus implement persistence homology algorithms pioneered in applied topology and data analysis by researchers at Stanford University and University of Illinois Urbana–Champaign. In quantum systems, computational homology aids in analyzing lattice models (e.g., the Kitaev model and Quantum error correction codes such as the Surface code), detecting topological order in condensed matter systems studied at Bell Labs and IBM Research, and in persistent homology analyses of quantum state manifolds in experiments at NIST and University of Oxford.
Cohomology theories, including de Rham cohomology and Čech cohomology, provide ring structures and products (cup and cap) that enrich homological data; these are essential in gauge theory and quantization. Homotopy-theoretic refinements, such as stable homotopy theory and spectra, lead to generalized homology theories like K-theory (utilized in classifying topological insulators by groups at University of Tokyo and Microsoft Research) and Morava K-theory in advanced field-theoretic contexts. Computational tools such as the Serre spectral sequence and Atiyah–Hirzebruch spectral sequence help compute homology of fibrations and filtered complexes; these techniques are used in the study of state spaces in quantum field theories and string compactifications investigated at Institute for Advanced Study and major university research groups.
Category:Algebraic topology Category:Mathematical physics