Cochains and homotopy type

By Michael A. Mandell. 2003

70 citations recorded by the source corpus.

Finite type nilpotent spaces are weakly equivalent if and only if their singular cochains are quasi-isomorphic as E∞ algebras. The cochain functor from the homotopy category of finite type nilpotent spaces to the homotopy category of E∞ algebras is faithful but not full.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3c83f5c4983d8df60fb2665defcf45141f1e5a03

How researchers see this paper

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In Algebra and Number Theory

1. G. Moore
2. D. Notbohm
3. Yankı Lekili
4. T. Bridgeland
5. Nigel Ray
6. A. Kapustin
7. Alastair Craw
8. J. Evans

In Geometry and Topology

1. P. Aspinwall
2. G. Moore
3. T. Bridgeland
4. Yukinobu Toda
5. A. Kapustin
6. G. Segal
7. Balázs Szendrői
8. Mark Gross

In Mathematical Physics

1. G. Moore
2. Yukinobu Toda
3. T. Bridgeland
4. P. Aspinwall
5. A. Kapustin
6. Michael R Douglas
7. G. Segal
8. Balázs Szendrői

Citations in the corpus

• Dirichlet Branes and Mirror Symmetry (2009, background): “A long evolution in algebraic topology has culminated in recent theorems of Mandell [346] which show that the actual homotopy category of topological spaces is more or less equivalent to the category of E∞-algebras.”
• Notes on factorization algebras, factorization homology and applications (2013, background): “ topological space. Its singular cochain complex C(X) has a natural structure of E ¥-algebra, whose underlying E 1-structure is given by the usual (strictly associative) cup-product (for instance see [M2]). The singular chains C(X) have a natural structure of E ¥-coalgebra which is the predual of (C”
• On Davis-Januszkiewicz homotopy types I; Formality and rationalisation (2003, background): “ld like to develop the results of Section 4 in the model category of E∞ algebras, which are emerging as the integral analogue of differential graded commutative algebras over Q. Recent work of Mandell [17] shows that c(K) is classified by the E∞ algebra C∗(c(K);Z), at least up to weak equivalence of n”
• D-branes and K-theory in 2D topological field theory (2006, unknown): “The differential in the complex is defined by (dF )(φ1, . . . , φk+1) =F (φ2, . . . , φk+1) ◦ φ1+ + k ∑ i=1 (−1)iF (φ1, . . . , φi+1 ◦ φi, . . . , φk+1) + (−1)k+1φk+1 ◦ F (φ1, . . . , φk).”
• Generating the Fukaya categories of Hamiltonian 𝐺-manifolds (2015, methodology): “By a theorem of Mandell [37], the mod p homotopy type of a simply-connected finite-type space X can be recovered from the E∞ - algebra structure on C∗(X; Fp).”