# A note on some homology spheres which are 2-fold coverings of

In the mathematical field of algebraic topologythe homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariantswhich reflect, in algebraic terms, the structure of spheres viewed as topological spacesforgetting about their precise geometry.

Unlike homology groupswhich 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 S n — generalizes the familiar circle S 1 and the ordinary sphere S 2. 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. These are called the stable homotopy groups of spheres and have been computed for values of k up to The stable homotopy groups form the coefficient ring of an extraordinary cohomology theorycalled stable cohomotopy theory. Most modern computations use spectral sequencesa technique first applied to homotopy groups of spheres by Jean-Pierre Serre.

Several important patterns have been established, yet much remains unknown and unexplained. The study of homotopy groups of spheres builds on a great deal of background material, here briefly reviewed. Algebraic topology provides the larger context, itself built on topology and abstract algebrawith homotopy groups as a basic example. An ordinary sphere in three-dimensional space — the surface, not the solid ball — is just one example of what a sphere means in topology.

Geometry defines a sphere rigidly, as a shape. Here are some alternatives. For some spaces the choice matters, but for a sphere all points are equivalent so the choice is a matter of convenience. The distinguishing feature of a topological space is its continuity structure, formalized in terms of open sets or neighborhoods.

A continuous map is a function between spaces that preserves continuity. A homotopy is a continuous path between continuous maps; two maps connected by a homotopy are said to be homotopic. The idea common to all these concepts is to discard variations that do not affect outcomes of interest.

An important practical example is the residue theorem of complex analysiswhere "closed curves" are continuous maps from the circle into the complex plane, and where two closed curves produce the same integral result if they are homotopic in the topological space consisting of the plane minus the points of singularity. These maps or equivalently, closed curves are grouped together into equivalence classes based on homotopy keeping the "base point" x fixedso that two maps are in the same class if they are homotopic.

The classes become an abstract algebraic group with the introduction of addition, defined via an "equator pinch". This pinch maps the equator of a pointed sphere here a circle to the distinguished point, producing a " bouquet of spheres " — two pointed spheres joined at their distinguished point.

The two maps to be added map the upper and lower spheres separately, agreeing on the distinguished point, and composition with the pinch gives the sum map. The null homotopic class acts as the identity of the group addition, and for X equal to S n for positive n — the homotopy groups of spheres — the groups are abelian and finitely generated.

A continuous map between two topological spaces induces a group homomorphism between the associated homotopy groups. In particular, if the map is a continuous bijection a homeomorphismso that the two spaces have the same topology, then their i -th homotopy groups are isomorphic for all i.

However, the real plane has exactly the same homotopy groups as a solitary point as does a Euclidean space of any dimensionand the real plane with a point removed has the same groups as a circle, so groups alone are not enough to distinguish spaces.

Although the loss of discrimination power is unfortunate, it can also make certain computations easier. The low-dimensional examples of homotopy groups of spheres provide a sense of the subject, because these special cases can be visualized in ordinary 3-dimensional space Hatcher However, such visualizations are not mathematical proofs, and do not capture the possible complexity of maps between spheres.

The simplest case concerns the ways that a circle 1-sphere can be wrapped around another circle. This can be visualized by wrapping a rubber band around one's finger: it can be wrapped once, twice, three times and so on. The wrapping can be in either of two directions, and wrappings in opposite directions will cancel out after a deformation. This integer can also be thought of as the winding number of a loop around the origin in the plane. Mappings from a 2-sphere to a 2-sphere can be visualized as wrapping a plastic bag around a ball and then sealing it.

The sealed bag is topologically equivalent to a 2-sphere, as is the surface of the ball.The definition implies that every covering map is a local homeomorphism. Covering spaces play an important role in homotopy theoryharmonic analysisRiemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group.

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In particular, covering maps are locally trivial. In particular, many authors require both spaces to be path-connected and locally path-connected. For every x in Xthe fiber over x is a discrete subset of C.

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If x and y are two points in X connected by a path, then that path furnishes a bijection between the fiber over x and the fiber over y via the lifting property. Since the unit interval [0, 1] is simply connected, the lifting property for paths is a special case of the lifting property for maps stated above.

Both of these statements can be deduced from the lifting property for continuous maps. Equivalence classes of coverings correspond to conjugacy classes of subgroups of the fundamental group of Xas discussed below. Since coverings are local homeomorphismsa covering of a topological n - manifold is an n -manifold. One can prove that the covering space is second-countable from the fact that the fundamental group of a manifold is always countable.

However a space covered by an n -manifold may be a non-Hausdorff manifold. The points p 1, 0 and p 0, 1 do not have disjoint neighborhoods in X. Any covering space of a differentiable manifold may be equipped with a natural differentiable structure that turns p the covering map in question into a local diffeomorphism — a map with constant rank n. A covering space is a universal covering space if it is simply connected. This can be phrased as. The space X has a universal cover if it is connectedlocally path-connected and semi-locally simply connected.

The universal cover of the space X can be constructed as a certain space of paths in the space X. The universal cover first arose in the theory of analytic functions as the natural domain of an analytic continuation. Let G be a discrete group acting on the topological space X.

Or in other words, a group action of the group G on the space X is just a group homomorphism of the group G into the group Homeo X of self-homeomorphisms of X. This is not always true since the action may have fixed points.

However the group G does act on the fundamental groupoid of Xand so the study is best handled by considering groups acting on groupoids, and the corresponding orbit groupoids.

The theory for this is set down in Chapter 11 of the book Topology and groupoids referred to below. This leads to explicit computations, for example of the fundamental group of the symmetric square of a space. Deck transformations are also called covering transformations. Every deck transformation permutes the elements of each fiber. This defines a group action of the deck transformation group on each fiber.That is.

It does not follow that X is simply connectedonly that its fundamental group is perfect see Hurewicz theorem. A rational homology sphere is defined similarly but using homology with rational coefficients. Being a spherical 3-manifoldit is the only homology 3-sphere besides the 3-sphere itself with a finite fundamental group.

Its fundamental group is known as the binary icosahedral group and has order A simple construction of this space begins with a dodecahedron. Each face of the dodecahedron is identified with its opposite face, using the minimal clockwise twist to line up the faces. Gluing each pair of opposite faces together using this identification yields a closed 3-manifold. See Seifert—Weber space for a similar construction, using more "twist", that results in a hyperbolic 3-manifold. One can also pass instead to the universal cover of SO 3 which can be realized as the group of unit quaternions and is homeomorphic to the 3-sphere.

Another approach is by Dehn surgery. If A is a homology 3-sphere not homeomorphic to the standard 3-sphere, then the suspension of A is an example of a 4-dimensional homology manifold that is not a topological manifold.

The double suspension of A is homeomorphic to the standard 5-sphere, but its triangulation induced by some triangulation of A is not a PL manifold.

In other words, this gives an example of a finite simplicial complex that is a topological manifold but not a PL manifold. It is not a PL manifold because the link of a point is not always a 4-sphere. As of [update] the existence of such a homology 3-sphere was an unsolved problem.

On March 11,Ciprian Manolescu posted a preprint on the ArXiv [5] claiming to show that there is no such homology sphere with the given property, and therefore, there are 5-manifolds not homeomorphic to simplicial complexes.

From Wikipedia, the free encyclopedia. Bibcode : Natur. Astronomy and Astrophysics. To appear in Journal of the AMS. Categories : Topological spaces Homology theory 3-manifolds Spheres. Hidden categories: Articles containing potentially dated statements from All articles containing potentially dated statements.The 2-spheredenotedis defined as the sphere of dimension 2. Below are some explicit definitions. The 2-sphere in with center and radius is defined as the following subset of :. In particular, the unit 2-sphere centered at the origin is defined as the following subset of :.

Note that all 2-spheres are equivalent up to translations and dilations, and in particular, they are homeomorphic as topological spaces. Further information: homology of spheres. The homology groups with coefficients in are as follows:and all other homology groups are zero. The reduced homology groups with coefficients in are as follows:and all other reduced homology groups are zero.

More generally, for homology with coefficients in any module over any commutative unital ringand all other homology groups are zero. For reduced homology,and all other reduced homology groups are zero. Further information: cohomology computation for spheres. The cohomology groups with coefficients in are as follows:and all other cohomology groups are zero.

The cohomology ring iswhere is an additive generator of. More generally, for coefficients in any commutative unital ring, and the other cohomology groups are zero. The cohomology ring iswhere is a generator of as a -module. Further information: homotopy of spheres. The 2-sphere is not a H-spacei. In particular, it does not arise from a topological monoid or a topological group. Further information: comultiplication on spheres.

The 2-sphere has a natural choice of comultiplication, i. This map is cocommutative and coassociative up to homotopy, and it is used to give an abelian group structure to the set of homotopy classes from the based 2-sphere to any based topological space. This group is termed the second homotopy group. Jump to: navigationsearch. This article is about a particular topological space uniquely determined up to homeomorphism View a complete list of particular topological spaces Contents.

Privacy policy About Topospaces Disclaimers Mobile view. Identification of antipodal points gives the double cover from to. Via stereographic projection, we see that the 2-sphere minus any point is homeomorphic to the Euclidean plane.

Thus, we can give it an atlas with two charts, each chart obtained by removing a different point and mapping homeomorphically to the Euclidean plane.

It is a union of two open subsets homeomorphic to the Euclidean plane hence path-connectedand with non-empty intersection. Thus, it is path-connected. Special case of n-sphere is simply connected for n greater than 1. Follows from union of two simply connected open subsets with path-connected intersection is simply connectedwhich is a corollary of the Seifert-van Kampen theorem.We consider finite groups G admitting orientation-preserving actions on homology 3-spheres arbitrary, i.

It is known that every finite group G admits actions on rational homology 3-spheres and even free actions. On the other hand, the class of groups admitting actions on integer homology 3-spheres is very restricted and close to the class of finite subgroups of the orthogonal group SO 4acting on the 3-sphere. From this we deduce a corresponding list for the case of integer homology 3-spheres. In the integer case, the groups of the list are closely related to the dodecahedral group or the binary dodecahedral group most of these groups are subgroups of the orthogonal group SO 4 and hence admit actions on S 3.

### Homotopy groups of spheres

This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Conway, J. Oxford University Press Aschbacher, M. Cambridge University Press Bredon, G. Academic Press, New York Brown, K. Graduate Texts in Mathematics 87, Springer Cooper, D. Dotzel, R. Du Val, P. Oxford Math.

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Monographs, Oxford University Press Gorenstein, D. Plenum Press, New York Hartley, R. Huppert, B. Berlin: Springer-Verlag Top Profile Members Research.

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Research Preprint M. Sakuma and Y. Yokota, An application of non-positively curved cubings of alternating linksarXiv Lee and M. Sakuma, A family of two generator non-Hopfian groupsarXiv Bowditch and M. Sakuma, The action of the mapping class group on the space of geodesic rays of a punctured hyperbolic surfacearXiv Publications D. Sakuma, Parabolic generating pairs of genus-one 2-bridge knot groupsJ.

Knot Theory Ramifications 25[21 pages], arXiv Sakuma, Simple loops on 2-bridge spheres in Heckoid orbifolds for the trivial knotEast Asian Math. Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in Heckoid orbifolds for 2-bridge links IJ. Knot Theory Ramifications 25[33 pages], arXiv Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in Heckoid orbifolds for 2-bridge links IIJ.

Knot Theory Ramifications 25[22 pages], arXiv Ohshika and M. Sakuma, Subgroups of mapping class groups related to Heegaard splittings and bridge decompositionsGeometria Dedicata Akiyoshi, D. Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements IGeom. Dedicata1— Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements IIGeom.

Dedicata29— Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements IIIGeom. Dedicata57— Sakuma, Simple loops on 2-bridge spheres in Heckoid orbifolds for 2-bridge linksElectron. Sakuma, Epimorphisms between 2-bridge link groups: homotopically trivial simple loops on 2-bridge spheresProc.

Sakuma, Simple loops on 2-bridge spheres in 2-bridge link complements. Dicks and M. Sakuma and K. Shackleton: On the distance between two Seifert surfaces of a knot Osaka Jour.

## Homotopy groups of spheres

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Akiyoshi and M. Sakuma, M. Wada, and Y. Yamshita, Punctured torus groups and 2-bridge knot groupsLecture Notes in Mathematics, Springer, Berlin, Sakuma, Epimorphisms between 2-bridge knot groups from the view point of markoff mapsIntelligence of Low Dimensional Topology Eds.

Scott Carter et.In the mathematical field of algebraic topologythe homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariantswhich reflect, in algebraic terms, the structure of spheres viewed as topological spacesforgetting about their precise geometry. Unlike homology groupswhich 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 S n — generalizes the familiar circle S 1 and the ordinary sphere S 2. 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.

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These are called the stable homotopy groups of spheres and have been computed for values of k up to The stable homotopy groups form the coefficient ring of an extraordinary cohomology theorycalled stable cohomotopy theory. Most modern computations use spectral sequencesa technique first applied to homotopy groups of spheres by Jean-Pierre Serre.

Several important patterns have been established, yet much remains unknown and unexplained. The study of homotopy groups of spheres builds on a great deal of background material, here briefly reviewed. Algebraic topology provides the larger context, itself built on topology and abstract algebrawith homotopy groups as a basic example. An ordinary sphere in three-dimensional space — the surface, not the solid ball — is just one example of what a sphere means in topology.

Geometry defines a sphere rigidly, as a shape. Here are some alternatives. For some spaces the choice matters, but for a sphere all points are equivalent so the choice is a matter of convenience. The distinguishing feature of a topological space is its continuity structure, formalized in terms of open sets or neighborhoods.

A continuous map is a function between spaces that preserves continuity. A homotopy is a continuous path between continuous maps; two maps connected by a homotopy are said to be homotopic. The idea common to all these concepts is to discard variations that do not affect outcomes of interest. An important practical example is the residue theorem of complex analysiswhere "closed curves" are continuous maps from the circle into the complex plane, and where two closed curves produce the same integral result if they are homotopic in the topological space consisting of the plane minus the points of singularity.

These maps or equivalently, closed curves are grouped together into equivalence classes based on homotopy keeping the "base point" x fixedso that two maps are in the same class if they are homotopic. The classes become an abstract algebraic group with the introduction of addition, defined via an "equator pinch". This pinch maps the equator of a pointed sphere here a circle to the distinguished point, producing a " bouquet of spheres " — two pointed spheres joined at their distinguished point.

The two maps to be added map the upper and lower spheres separately, agreeing on the distinguished point, and composition with the pinch gives the sum map.