Related papers: Rational obstruction theory and rational homotopy …

We describe algebraic obstruction theories for realizing an abstract coalgebra K_* over the mod p Steenrod algebra as the homology of a topological space, and for distinguishing between the p-homotopy types of different realizations. The…

The space of E-infinity structures on an simplicial operad C is the limit of a tower of fibrations, so its homotopy is the abutment of a Bousfield-Kan fringed spectral sequence. The spectral sequence begins (under mild restrictions) with…

The Lefschetz fixed point theorem provides a powerful obstruction to the existence of minimal homeomorphisms on well-behaved spaces such as finite CW-complexes. We show that these obstructions do not hold for more general spaces. More…

In this paper, we show that for finite $CW$-complexes $X$ and two-stage space $Y$ (for example $n$-spheres $S^n$, homogeneous spaces and $F_0$-spaces), the rational homotopy type of $\map(X, Y)$ is determined by the cohomology algebra…

Given a diagram of Pi-algebras (graded groups equipped with an action of the primary homotopy operations), we ask whether it can be realized as the homotopy groups of a diagram of spaces. The answer given here is in the form of an…

Using the notion of isotopy modulo $k$, with $k \in \mathbb{N}^+$, we introduce a stratification on the set of all minimal $C_\infty$-algebra enhancements of a finite-type graded commutative algebra $H^*$. We determine obstruction classes…

We consider a homotopy theory obtained from that of pointed spaces by inverting the maps inducing isomorphisms in $v_n$-periodic homotopy groups. The case n = 0 corresponds to rational homotopy theory. In analogy with Quillen's results in…

We give a construction of the obstruction theory for $\mathbb{A}_{n}$-algebra structures in stable $\infty$-categories, and give some properties of it. We use this to show that the spectrum $\mathbb{S} / 4$ admits an…

Let $f:A \to B$ be a ring homomorphism of not necessarily unital rings and $I\triangleleft A$ an ideal which is mapped by f isomorphically to an ideal of B. The obstruction to excision in K-theory is the failure of the map between relative…

We develop a homotopical variant of the classic notion of an algebraic theory as a tool for producing deformations of homotopy theories. From this, we extract a framework for constructing and reasoning with obstruction theories and spectral…

We investigate various homotopy invariant formulations of commutative algebra in the context of rational homotopy theory. The main subject is the complete intersection condition, where we show that a growth condition implies a structure…

We develop an obstruction theory for Hirsch extensions of cbba's with twisted coefficients. This leads to a variety of applications, including a structural theorem for minimal cbba's, a construction of relative minimal models with twisted…

If $M$ is an $m \times m$ matrix over $\{ 0, 1, \ast \}$, an $M$-partition of a graph $G$ is a partition $(V_1, \dots V_m)$ such that $V_i$ is completely adjacent (non-adjacent) to $V_j$ if $M_{ij} = 1$ ($M_{ij} = 0$), and there are no…

After introducing some cohomology classes as obstructions to orientation and spin structures etc., we explain some applications of cohomology to physical problems, in special to reduced holonomy in M- and F-theory.

We develop a general obstruction theory to the formality of algebraic structures over any commutative ground ring. It relies on the construction of Kaledin obstruction classes that faithfully detect the formality of differential graded…

Given a topological group G, its orbit category Orb_G has the transitive G-spaces G/H as objects and the G-equivariant maps between them as morphisms. A well known theorem of Elmendorf then states that the category of G-spaces and the…

In this note, we prove an obstruction theorem for the existence of A infinite-structures over a commutative ring R on an algebra A associative up to homotopy, in terms of the Hochschild cohomology of the associative algebra H(A). The hidden…

We consider the following problem for a fixed graph H: given a graph G and two H-colorings of G, i.e. homomorphisms from G to H, can one be transformed (reconfigured) into the other by changing one color at a time, maintaining an H-coloring…

A series of recent papers by Bergfalk, Lupini and Panagiotopoulus developed the foundations of a field known as `definable algebraic topology,' in which classical cohomological invariants are enriched by viewing them as groups with a Polish…

We obtain restrictions on the rational homotopy types of mapping spaces and of classifying spaces of homotopy automorphisms by means of the theory of positive weight decompositions. The theory applies, in particular, to connected components…