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We develop a machinery of Chen iterated integrals for higher Hochschild complexes. These are complexes whose differentials are modeled on an arbitrary simplicial set much in the same way the ordinary Hochschild differential is modeled on the circle. We use these to give algebraic models for general mapping spaces and define and study the surface product operation on the homology of mapping spaces of surfaces of all genera into a manifold. This is an analogue of the loop product in string topology....

A classification of small homotopy functors from spectra to spectra

Boris Chorny (2016)

Fundamenta Mathematicae

We show that every small homotopy functor from spectra to spectra is weakly equivalent to a filtered colimit of representable functors represented in cofibrant spectra. Moreover, we present this classification as a Quillen equivalence of the category of small functors from spectra to spectra equipped with the homotopy model structure and the opposite of the pro-category of spectra with the strict model structure.

A Coclassifying Map for the Inclusion of the Wedge in the Product.

John W. Rutter (1972)

Mathematische Zeitschrift

A Cohomological Proof of the Torus Theorem.

James P. Lin (1985)

Mathematische Zeitschrift

A comparison principle and extension of equivariant maps.

Alexander Kushkuley, Zalman Balanov (1994)

Manuscripta mathematica

A convenient category for directed homotopy.

Fajstrup, L., Rosický, J. (2008)

Theory and Applications of Categories [electronic only]

A counterexample concerning products in the shape category

J. Dydak, S. Mardešić (2005)

Fundamenta Mathematicae

We exhibit a metric continuum X and a polyhedron P such that the Cartesian product X × P fails to be the product of X and P in the shape category of topological spaces.

A Counterexample to a Conjecture of Steenrod.

Gunnar Carlsson (1981)

Inventiones mathematicae

A criterion of SNT(X) = {[X]} for hyperformal spaces

Jinsong Ni (2009)

Open Mathematics

We will give a condition characterizing spaces X with SNT(X) = {[X]} which generalizes the corresponding result of McGibbon and Moller [8] for rational H-spaces.

A Filtration of the Loops on SU (N) by Schubert Varieties.

Steven A. Mitchell (1986)

Mathematische Zeitschrift

A formula for the rational LS-category of certain spaces

Luis Lechuga, Aniceto Murillo (2002)

Annales de l’institut Fourier

In this paper we find a formula for the rational LS-category of certain elliptic spaces which generalizes or complements previous work of the subject. This formula is given in terms of the minimal model of the space.

A formula for topology/deformations and its significance

Ruth Lawrence, Dennis Sullivan (2014)

Fundamenta Mathematicae

The formula is $\partial e = (a d_{e}) b + \sum_{i = 0}^{\infty} (B_{i}) / i! {(a d_{e})}^{i} (b - a)$ , with ∂a + 1/2 [a,a] = 0 and ∂b + 1/2 [b,b] = 0, where a, b and e in degrees -1, -1 and 0 are the free generators of a completed free graded Lie algebra L[a,b,e]. The coefficients are defined by $x / (e^{x} - 1) = \sum_{n = 0}^{\infty} B ₙ / n! x ⁿ$ . The theorem is that ∙ this formula for ∂ on generators extends to a derivation of square zero on L[a,b,e]; ∙ the formula for ∂e is unique satisfying the first property, once given the formulae for ∂a and ∂b, along with the condition that the “flow” generated by e moves a to b in unit...

A functional S-dual in a strong shape category

Friedrich Bauer (1997)

Fundamenta Mathematicae

In the S-category $P$ (with compact-open strong shape mappings, cf. §1, instead of continuous mappings, and arbitrary finite-dimensional separable metrizable spaces instead of finite polyhedra) there exists according to [1], [2] an S-duality. The S-dual $D X, X = (X, n) \in P$ , turns out to be of the same weak homotopy type as an appropriately defined functional dual $\bar{{(S^{0})}^{X}}$ (Corollary 4.9). Sometimes the functional object $\bar{X^{Y}}$ is of the same weak homotopy type as the “real” function space $X^{Y}$ (§5).

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