Loop Spaces and Higher Homotopy Groups

We study the etale homotopy theory of Brauer-Severi varieties over fields of characteristic 0. We prove that the induced Galois representations on geometric homotopy invariants (e.g., l-adic cohomology or higher homotopy groups) are all isomorphic for Brauer-Severi varieties of the same dimension. If the base field has cohomological dimension smaller or equal 2 then we can show more in the case of Brauer-Severi curves: There is even an isomorphism between the Hochschild-Serre spectral sequences computing cohomology with local coefficients. Further, we study homotopy rational and homotopy fixed points on Brauer-Severi varieties and their connections to genuine rational points. In particular, we show that under a suitable assumption on the first profinite Chern class map an analogue of the weak section conjecture for Brauer-Severi varieties turns out to be true. We can give a counter example to this analogue without the extra assumption over p-adic local fields.

The infinitary n-cube shuffle

Rational homotopy theory and differential graded category

In this paper, we discuss sections of higher homotopy groupoid bundles those yield by group actions. We study some basic properties related to the left-invariant (G-invariant) of such sections and also present some conditions for such sections to be left-invariant (G-invariant). We concentrate a kind of group action on the Core higher homotopy groupoid bundle of certain topological spaces which undergo a regular topological group action. Interestingly, each element of orbit space yields continuous sections of the Core higher homotopy groupoid bundle having beautiful algebraic properties. Moreover, each of those continuous sections behaves nicely as a left-invariant section with respect to topological group action by G. Since higher homotopy groups are abelian, the set of such sections forms always an abelian group. Moreover, each element of the group is a left-invariant section, therefore, it is convenient to call them commutative group of left-invariant sections of the Core higher homotopy groupoid bundle. Further, we show this group is a topological invariant and study some basic properties of it. At the end of the paper, we give interrelations between the retraction map and this group.

where h* denotes the induced homomorphism. Note that #* is an isomorphism if A is a covering map and p, a ^ 2 . Hence if two topological groups have a common universal covering group then their higher homotopy groups are related by an isomorphism which is compatible with the Samelson product. Let <nrq(G), where g ^ l , denote the subset of w2q(G) consisting of elements ((3, fi), where fiCz7Tq(G). We assert the following

A proof of a weak form of Serre's theorem on the finiteness of stable homotopy groups of spheres is suggested, which uses neither loop spaces nor spectral sequences.

If q: X → Y is a quotient map, then, in general, q × q: X × X → Y × Y may fail to be a quotient map. In this paper, by reviewing the concept of homotopy groups and quotient maps, we find under which conditions the map q × q is a quotient map, where q: Ω n (X, x 0) → πn (X, x 0), is the natural quotient map from the nth loop space of (X, x 0), Ω n (X, x 0), with compact-open topology to the quasitopological nth homotopy group πn (X, x 0). Ultimately, using these results, we found some properties of first countable homotopy groups.

Some examples of homotopy Π-algebras

This note explores J. H. C. Whitehead’s 1941 question as to whether a subcomplex of an aspherical 2-complex need also have vanishing higher homotopy groups. Methods from \(L^{2}\)-cohomology are brought to bear on the question and relate it to other open problems on low-dimensional complexes—some introduced here—as well as open problems on group theory, such as the Kervaire–Laudenbach Conjecture, and on group algebras, like the Bass trace conjecture.

We study localization at a prime in homotopy type theory, using self maps of the circle. Our main result is that for a pointed, simply connected type $X$, the natural map $X \to X_{(p)}$ induces algebraic localizations on all homotopy groups. In order to prove this, we further develop the theory of reflective subuniverses. In particular, we show that for any reflective subuniverse $L$, the subuniverse of $L$-separated types is again a reflective subuniverse, which we call $L'$. Furthermore, we prove results establishing that $L'$ is almost left exact. We next focus on localization with respect to a map, giving results on preservation of coproducts and connectivity. We also study how such localizations interact with other reflective subuniverses and orthogonal factorization systems. As key steps towards proving the main theorem, we show that localization at a prime commutes with taking loop spaces for a pointed, simply connected type, and explicitly describe the localization of an Eilenberg-Mac Lane space $K(G,n)$ with $G$ abelian. We also include a partial converse to the main theorem.

The root invariant in homotopy theory

We establish a pseudoisotopy result for embedding spaces in the line of that of Weiss and Williams for diffeomorphism groups. In other words, for $P\subset M$ a codimension at least three embedding, we describe the difference in a range of homotopical degrees between the spaces of block and ordinary embeddings of $P$ into $M$ as a certain infinite loop space involving the relative algebraic $K$-theory of the pair $(M,M-P)$. This range of degrees is the so-called concordance embedding stable range, which, by recent developments of Goodwillie-Krannich-Kupers, is far beyond that of the aforementioned theorem of Weiss-Williams. We use this result to obtain split fibre sequences in the concordance embedding stable range, with explicit, analysable base and fibre, which determine the homotopy type of spaces of long knots of codimension at least 3. This leads to explicit computations of homotopy groups, including torsion information, in that range. In doing so, we carry out an extensive analysis of certain geometric involutions in algebraic $K$-theory that may be of independent interest.

Chapter 24 - Fibration and Product Decompositions in Nonstable Homotopy Theory

If X is a space with base point, let ΩX denote the space of loops of X based at its base point. We will denote the homology and cohomology suspensions by σ* and σ*, respectively, , . Let K(π, n) be the Eilenberg-MacLane complex with only one non-vanishing homotopy group πn(K(π, n))= π, πi(K(π, n)) = 0 for i ≠ n. If π is abelian K(π, n) = ΩK(π, n+1). We will assume that π is a finitely generated abelian group throughout this note. Let K(π, n) × K(π, n) → K(π, n) denote the multiplication in K(π, n).

Let [math] be a Poincaré duality space, [math] a space and [math] a based map. We show that the rational homotopy group of the connected component of the space of maps from [math] to [math] containing [math] is contained in the rational homology group of a space [math] which is the pullback of [math] and the evaluation map from the free loop space [math] to the space [math] . As an application of the result, when [math] is a closed oriented manifold, we give a condition of a noncommutativity for the rational loop homology algebra [math] defined by Gruher and Salvatore which is the extension of the Chas–Sullivan loop homology algebra.