Postnikov Systems. On Mappings into Group-like Spaces. Homotopy Operations. Stable Homotopy and Homology. Homology of Fibre Spaces. Back Matter Pages About this book Introduction As the title suggests, this book is concerned with the elementary portion of the subject of homotopy theory.
It is assumed that the reader is familiar with the fundamental group and with singular homology theory, including the Universal Coefficient and Kiinneth Theorems. Some acquaintance with manifolds and Poincare duality is desirable, but not essential. Anyone who has taught a course in algebraic topology is familiar with the fact that a formidable amount of technical machinery must be introduced and mastered before the simplest applications can be made. This phenomenon is also observable in the more advanced parts of the subject.
I have attempted to short-circuit it by making maximal use of elementary methods. This approach entails a leisurely exposition in which brevity and perhaps elegance are sacrificed in favor of concreteness and ease of application. It is my hope that this approach will make homotopy theory accessible to workers in a wide range of other subjects-subjects in which its impact is beginning to be felt. It is a consequence of this approach that the order of development is to a certain extent historical.
Indeed, if the order in which the results presented here does not strictly correspond to that in which they were discovered, it nevertheless does correspond to an order in which they might have been discovered had those of us who were working in the area been a little more perspicacious. They consequently exhibit a Galois symmetry on their profinite completions.
Example 1 for any comes from a variety over even. In particular, the colimit has a Galois symmetry: the action on of an element is given by multiplication by. This follows from the computation we did earlier for and naturality thanks to the inclusion. For , and for the following example, the Galois symmetry is not at all evident. Example 2 Each Grassmannian of -planes in -space in fact, these exist as schemes over.
Taking the colimit in , we get a Galois symmetry on. Using the splitting principle for the -algebraic! Since the maps come from maps of -varieties over , the Galois action on the profinite completions is compatible with these inclusion maps. Our goal in this section is to use the Galois symmetry in the profinite completions of to produce the promised unstable versions of the Adams operations. These will be maps. To do this, choose whose cyclotomic character projects to.
As in the last section, we get a map. These maps are compatible in for varying , and has the action in -cohomology. Proposition 4 The map is an unstable version of the Adams operations. To do this, we have to show that two elements in. For this in turn it suffices by the splitting principle for -theory to pull back along the map. In other words, we need to show that the following diagram is homotopy commutative:.
The Adams conjecture is now a corollary of the above analysis. Namely, as above, we were reduced to showing that the two maps. Now the first map for classifies the tautological spherical fibration. The second map, when we twist by , classifies the pull-back of this fibration along. But the homotopy commutative and cartesian!
This is the Adams conjecture. You are commenting using your WordPress. You are commenting using your Google account. You are commenting using your Twitter account. You are commenting using your Facebook account. Notify me of new comments via email. Notify me of new posts via email. Blog at WordPress. January 25, Genetics of homotopy theory and the Adams conjecture Posted by Akhil Mathew under algebraic geometry , topology Tags: Adams conjecture , etale cohomology , etale homotopy , genetics of homotopy theory Leave a Comment.
Our goal is to show that the composite is nullhomotopic. Since the homotopy groups of are finite, it will follow by the Milnor exact sequence that we can let and conclude that the map is nullhomotopic i. Using again the finiteness of the homotopy groups of , we can get a splitting into the respective profinite completions.
It suffices therefore to prove that for a prime , the map is nullhomotopic. Here is the space that classifies families of -adically completed copies of the sphere spectrum; it has the property that but the higher homotopy groups are that of. The map classifies the completed spherical fibration over associated to fibration given by -completing the inclusion map. The key observation of Sullivan that leads to a proof of the Adams conjecture is: Theorem 1 Suppose.
Then the map canonically factors through , giving an unstable Adams operation which is a homotopy equivalence. In particular, we have a commutative diagram The Adams conjecture is in fact a consequence of these observations: It shows that the spherical fibration over classified by is exactly the pull-back of along.
The above diagram and the fact that the horizontal maps are homotopy equivalences! Let be the functor that sends a pro-object to its homotopy limit. The important is the following: Theorem 2 Let be a variety defined over. It follows that one has an isomorphism, -equivariant, Thus we have a -equivariant isomorphism Concretely, this means the following: there is a cyclotomic character given by restriction to the maximal cyclotomic extension.
We find: Proposition 3 The action of on induces multiplication by on with finite or profinite coefficients. More examples of Galois symmetry Many other natural examples of spaces come from varieties over or -varieties. Taking the colimit in , we get a Galois symmetry on Using the splitting principle for the -algebraic! The profinite Adams operations Our goal in this section is to use the Galois symmetry in the profinite completions of to produce the promised unstable versions of the Adams operations.
These will be maps for which are self-homotopy equivalences and which are compatible with the usual profinitely completed Adams operations. As in the last section, we get a map These maps are compatible in for varying , and has the action in -cohomology in particular, it is an equivalence since we are completed at. Proof: We need to show that the diagram is homotopy commutative, where is the ordinary Adams operation.
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