Motivic Homotopy Theory

... not enough since the structural sheaf is also part of the structure of a scheme. Moreover, given a homotopy theory of schemes, how to construct cohomology theories ? An answer to the first question is given by the machinery invented by Quillen, the model categories, introduced in [Qui67]. A model st ...

... not enough since the structural sheaf is also part of the structure of a scheme. Moreover, given a homotopy theory of schemes, how to construct cohomology theories ? An answer to the first question is given by the machinery invented by Quillen, the model categories, introduced in [Qui67]. A model st ...

The Simplicial Lusternik

... The simplicial Lusternik-Schnirelmann category is the definition of LusternikSchnirelmann category for simplicial complexes. The Lusternik-Schnirelmann (LS) category of a topological space X represents the least number n such that there is an open cover of X of n + 1 subsets contractible to a point ...

... The simplicial Lusternik-Schnirelmann category is the definition of LusternikSchnirelmann category for simplicial complexes. The Lusternik-Schnirelmann (LS) category of a topological space X represents the least number n such that there is an open cover of X of n + 1 subsets contractible to a point ...

SYMMETRIC SPECTRA Contents Introduction 2 1

... a symmetric monoidal category of spectra, called S-modules. Some generalizations of symmetric spectra appear in [MMSS1]. These many new symmetric monoidal categories of spectra, including S-modules and symmetric spectra, are shown to be equivalent in an appropriate sense in [MMSS1] and [MMSS2]. Anot ...

... a symmetric monoidal category of spectra, called S-modules. Some generalizations of symmetric spectra appear in [MMSS1]. These many new symmetric monoidal categories of spectra, including S-modules and symmetric spectra, are shown to be equivalent in an appropriate sense in [MMSS1] and [MMSS2]. Anot ...

on the ubiquity of simplicial objects

... In this thesis we introduce simplicial objects and then treat four different areas in which they are used: Homotopy theory, homological algebra, higher category theory and cohomological descent. The first chapter is a short introduction to the basic definitions of simplicial objects in arbitrary cat ...

... In this thesis we introduce simplicial objects and then treat four different areas in which they are used: Homotopy theory, homological algebra, higher category theory and cohomological descent. The first chapter is a short introduction to the basic definitions of simplicial objects in arbitrary cat ...

CW-complexes in the category of exterior spaces

... In this work we do a deeper study of the category of exterior spaces by developing a theory of exterior CW-complexes. This study will give several interesting consequences in proper homotopy. Among these results we can mention Proper Whitehead Theorem Proper Cellular Approximation Theorem (they may ...

... In this work we do a deeper study of the category of exterior spaces by developing a theory of exterior CW-complexes. This study will give several interesting consequences in proper homotopy. Among these results we can mention Proper Whitehead Theorem Proper Cellular Approximation Theorem (they may ...

as a PDF - Universität Bonn

... oriented smooth surface bundles; and the contractibility of S(M ) shows that on any surface bundle E → B, we can find complex structures on the fibers which vary continuously. Moreover, all these complex structures are concordant. The next important and much deeper step towards the homotopical theor ...

... oriented smooth surface bundles; and the contractibility of S(M ) shows that on any surface bundle E → B, we can find complex structures on the fibers which vary continuously. Moreover, all these complex structures are concordant. The next important and much deeper step towards the homotopical theor ...

Sheaves of Groups and Rings

... Remark 1. Let X, P be as in Definition 3. Then P is a sheaf if and only if for every open set U and indexed open cover (I, c) of U there is a unique amalgamation of every indexed matching family for the cover (I, c) and presheaf P . Remark 2. In practice, the following characterisation is more conv ...

... Remark 1. Let X, P be as in Definition 3. Then P is a sheaf if and only if for every open set U and indexed open cover (I, c) of U there is a unique amalgamation of every indexed matching family for the cover (I, c) and presheaf P . Remark 2. In practice, the following characterisation is more conv ...

Algebraic Topology

... To avoid overusing the word ‘continuous’ we adopt the convention that maps between spaces are always assumed to be continuous unless otherwise stated. ...

... To avoid overusing the word ‘continuous’ we adopt the convention that maps between spaces are always assumed to be continuous unless otherwise stated. ...

Lecture Notes

... Then we say that (X, F ) is a topological space whose open sets are the elements of F . We say F is the topology on X. Note we are choosing the collection F of open sets for X. There is more than one choice of F for a given X. Just remember balls are not defined in an arbitrary topological space. Ex ...

... Then we say that (X, F ) is a topological space whose open sets are the elements of F . We say F is the topology on X. Note we are choosing the collection F of open sets for X. There is more than one choice of F for a given X. Just remember balls are not defined in an arbitrary topological space. Ex ...

Topolog´ıa Algebraica de Espacios Topológicos Finitos y Aplicaciones

... between simple homotopy types of finite spaces and of simplicial complexes. This fundamental result allows us to study well-known geometrical problems from a new point of view, using all the combinatorial and topological machinery proper of finite spaces. Quillen’s conjecture on the poset of p-subgr ...

... between simple homotopy types of finite spaces and of simplicial complexes. This fundamental result allows us to study well-known geometrical problems from a new point of view, using all the combinatorial and topological machinery proper of finite spaces. Quillen’s conjecture on the poset of p-subgr ...

Surveys on Surgery Theory : Volume 1 Papers dedicated to C. T. C.

... the more general problem of classifying (n − 1)-connected 2n-dimensional manifolds, for all n. The intersection form on middle homology again appears as a fundamental algebraic invariant of oriented homotopy type. In fact this invariant also makes sense for an (n−1)-connected 2n-dimensional manifold ...

... the more general problem of classifying (n − 1)-connected 2n-dimensional manifolds, for all n. The intersection form on middle homology again appears as a fundamental algebraic invariant of oriented homotopy type. In fact this invariant also makes sense for an (n−1)-connected 2n-dimensional manifold ...

MAPPING STACKS OF TOPOLOGICAL STACKS Contents 1

... where BG is the Milnor classifying space of G. 2.3. Hurewicz topological stacks. One drawback of the 2-category of topological stacks is that homotopy between maps of topological stacks is not in general an equivalence relation (because it may not be transitive). More precisely, homotopies between m ...

... where BG is the Milnor classifying space of G. 2.3. Hurewicz topological stacks. One drawback of the 2-category of topological stacks is that homotopy between maps of topological stacks is not in general an equivalence relation (because it may not be transitive). More precisely, homotopies between m ...

Topological constructors

... Algebraic topology is concerned by any kind of topological space, but using the algebraic tool soon leads to favour the constructors producing spaces which can be conveniently so analyzed. Two important constructors are considered here, the first one giving the CW-complexes, the second one the simpl ...

... Algebraic topology is concerned by any kind of topological space, but using the algebraic tool soon leads to favour the constructors producing spaces which can be conveniently so analyzed. Two important constructors are considered here, the first one giving the CW-complexes, the second one the simpl ...

E∞-Comodules and Topological Manifolds A Dissertation presented

... in the cohomology of a triangulated space, which is associative and graded commutative, can be induced from a cochain level product satisfying the same two properties. He answered it in the negative after identifying homological obstructions among a collection of chain maps he constructed. Using lat ...

... in the cohomology of a triangulated space, which is associative and graded commutative, can be induced from a cochain level product satisfying the same two properties. He answered it in the negative after identifying homological obstructions among a collection of chain maps he constructed. Using lat ...

# Homotopy groups of spheres

In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute.The n-dimensional unit sphere — called the n-sphere for brevity, and denoted as Sn — generalizes the familiar circle (S1) and the ordinary sphere (S2). The n-sphere may be defined geometrically as the set of points in a Euclidean space of dimension n + 1 located at a unit distance from the origin. The i-th homotopy group πi(Sn) summarizes the different ways in which the i-dimensional sphere Si can be mapped continuously into the n-dimensional sphere Sn. This summary does not distinguish between two mappings if one can be continuously deformed to the other; thus, only equivalence classes of mappings are summarized. An ""addition"" operation defined on these equivalence classes makes the set of equivalence classes into an abelian group.The problem of determining πi(Sn) falls into three regimes, depending on whether i is less than, equal to, or greater than n. For 0 < i < n, any mapping from Si to Sn is homotopic (i.e., continuously deformable) to a constant mapping, i.e., a mapping that maps all of Si to a single point of Sn. When i = n, every map from Sn to itself has a degree that measures how many times the sphere is wrapped around itself. This degree identifies πn(Sn) with the group of integers under addition. For example, every point on a circle can be mapped continuously onto a point of another circle; as the first point is moved around the first circle, the second point may cycle several times around the second circle, depending on the particular mapping. However, the most interesting and surprising results occur when i > n. The first such surprise was the discovery of a mapping called the Hopf fibration, which wraps the 3-sphere S3 around the usual sphere S2 in a non-trivial fashion, and so is not equivalent to a one-point mapping.The question of computing the homotopy group πn+k(Sn) for positive k turned out to be a central question in algebraic topology that has contributed to development of many of its fundamental techniques and has served as a stimulating focus of research. One of the main discoveries is that the homotopy groups πn+k(Sn) are independent of n for n ≥ k + 2. These are called the stable homotopy groups of spheres and have been computed for values of k up to 64. The stable homotopy groups form the coefficient ring of an extraordinary cohomology theory, called stable cohomotopy theory. The unstable homotopy groups (for n < k + 2) are more erratic; nevertheless, they have been tabulated for k < 20. Most modern computations use spectral sequences, a technique first applied to homotopy groups of spheres by Jean-Pierre Serre. Several important patterns have been established, yet much remains unknown and unexplained.