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Representability of Cohomology Theories on CW-Complexes

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Representability of Cohomology Theories on CW-Complexes Representability of Cohomology Theories on CW-complexes Bachelor's Project Mathematics 09-07-2021 Student: D. Apol First supervisor: Dr. R.I. van der Veen Second assessor: Dr. O. Lorscheid Daily supervisor: Drs. J. Becerra Abstract In topology, it is in general very difficult to decide whether or not two spaces are homotopy equivalent. In algebraic topology, therefore, algebraic invariants are associated with each space, reflecting some topological data. Examples of this are homology and cohomology, that originated from measuring the number and size of holes in topological spaces. They turn out to be versatile and related to much that is cared about in algebraic topology. In this thesis, we will show that, as long as we restrict ourselves to a certain subclass of topological spaces, namely that of CW-complexes, all cohomology theories are representable by a sequence of CW-complexes. To do so, we first explore singular homology and cohomology, define what generalised cohomology theories are, and use the homotopy theory of CW-complexes to prove this main result. i Contents Introduction iii Conventions................................................... iii Notation..................................................... iv 1 Preliminaries 1 1.1 Singular homology............................................1 1.2 Relative homology............................................6 1.3 Properties of singular homology.....................................7 2 Singular cohomology 11 2.1 Dualising singular homology....................................... 11 2.2 Reduced singular cohomology...................................... 15 3 CW-complexes 19 3.1 Pushout diagrams............................................ 19 3.2 Construction of CW-complexes..................................... 22 3.3 Cones, suspensions, mapping cones and mapping cylinders...................... 25 4 Homotopy theory of CW-complexes 29 4.1 The homotopy extension property................................... 29 4.2 Higher homotopy groups......................................... 36 4.3 Approximation theorems and the Whitehead Theorem........................ 40 5 Cohomology on CW-complexes 45 5.1 Reduced and generalised cohomology theories on CW-complexes.................. 45 5.2 The equivalence of categories...................................... 48 5.3 Interlude on semi-Brown functors.................................... 52 6 Ω-spectra and representability of cohomology 57 6.1 Mapping spaces and the suspension-loop adjunction......................... 57 6.2 Ω-spectra................................................. 59 6.3 The Brown Representability Theorem................................. 63 6.4 Eilenberg-MacLane spaces........................................ 70 6.5 Two further examples of cohomology theories............................. 75 Discussion and conclusion 77 A Category theory 79 A.1 Categories and functors......................................... 79 A.2 Natural transformations and the Yoneda Lemma........................... 82 A.3 Adjunctions, limits and colimits..................................... 84 B Homological algebra 88 B.1 Exact sequences of modules....................................... 88 B.2 Chain complexes and homological algebra............................... 90 B.3 Dual cochain complexes and the Algebraic Universal Coefficient Theorem............. 95 ii Introduction A central challenge in topology is to show if two given topological space X and Y are homeomorphic or homotopy equivalent. In practice, there is often not a natural candidate for a homeomorphism or homotopy equivalence, and to directly show no such map exists is neigh impossible. Therefore, algebraic topology seeks to assign to each topological space X some algebraic structure that is invariant under homeomorphism or homotopy equivalence. If two spaces X and Y have non-isomorphic invariants associated to them, consequently they cannot be homeomorphic or homotopy equivalent, respectively. It is in practice doable to check whether or not groups or modules (or whatever the algebraic structure is) are isomorphic, so this is a viable approach. The fundamental group π1(X; x) is one example of such an invariant, and measures the amount of different loops there are in X on a certain base point x (up to homotopy). As such, it is also a measure of the amount of 2 holes in the space, although not a very reliable one in that. For instance, we know that π1(S ) = 0, although we would certainly say the 2-sphere has a two-dimensional hole (since the boundary of this hole is a surface). Apparently, the fundamental group cannot recognise this, so it does not capture all the information we are after. If we want to define an algebraic invariant on an arbitrary space, it is natural to want to measure the amount and dimension of the holes of such a space. They are namely not only visibly homotopy invariants on the space, but are themselves visible: they are clear geometric aspects of the space, one of the few we have left once we make the step from more rigid geometry like differential geometry to topology. This leads us to singular homology and cohomology, which we will be covering in the first two chapters of this thesis. These assign for each n ≥ 0 a module over a ring to a given space. The most important properties of singular cohomology can be used as a starting point to define what we wish a cohomology theory in general to be. We then do not have an explicit description of the inner workings of the cohomology theory anymore, as we do have with singular cohomology. What we will however be concerned with showing is that despite this, all cohomology theories that map to abelian groups (which are Z-modules) actually arise in a similar manner, as long as we restrict ourselves to CW-complexes only: for any cohomology theory, there exist fixed CW-complexes such that the algebraic invariants that the theory assigns to a CW-complex X is the set of homotopy classes of maps from X into those fixed spaces (and we will also see that this set carries a group structure). This is of huge theoretical and practical importance, we will encounter some beautiful consequences of it along the way. This thesis is meant as a Bachelor's level treatment of cohomology, CW-complexes, homotopy theory and representability of said cohomology theories. As such, not much prior knowledge is required. A solid under- standing of point-set topology and of group and module theory are needed, and a basic understanding the fundamental group and homotopy are recommended. Although we do include some category theory, module theory and homological algebra in the appendices, we omit most of the proofs of the statements there, and it certainly helps if the reader is already a bit familiar with those topics. Conventions We quickly write down a few conventions that we adopt in this thesis, that are not entirely standard. − Every ring is understood to have a unit. − The empty set ? is also considered a topological space because of category theoretical considerations. However, arbitrary topological spaces are often implicitly assumed to be nonempty in the text. This is because arguments with maps to and from empty spaces require some care, although it is always simple to check if a certain passage still holds for the empty space, and to check that results hold for empty spaces as well. It is therefore not worth the effort to always write down a separate passage for the empty space. − A space X is quasi-compact if every open cover of X admits a finite subcover. A space X is compact if it is both quasi-compact and Hausdorff. − A space X is locally compact if for every point x 2 X and open neighbourhood U ⊆ X, there is a compact (in the sense of the above convention) neighbourhood K ⊆ X of x with K ⊆ U. K being a neighbourhood of x means that there also exists an open O 3 x with O ⊆ K. 0 0 − If X ⊆ X are two topological spaces, a map f : X ! X is a deformation retraction if fjX0 = idX0 and 0 if there exists a homotopy H : X × [0; 1] ! X from idX to ι ◦ f, where ι : X ,! X is the inclusion, such that H( · ; t)jX0 = idX0 for all t. iii Notation Ab The category of abelian groups. Example A.4. Cf,Cuf The reduced and unreduced mapping cone of a map. Definition 3.45. CX, CuX The reduced and unreduced cone on X. Definition 3.42. C∗ The pointed version of a category C. C(2), C(3) The category of pairs or triplets of objects in C. Copp The opposite category of a category C. Definition A.8. RChain, RcChain The category of left R-module chain or cochain complexes. Notation B.20. C•(X; M) The singular chain complex with coefficients in M. Definition 1.15. C•(X; M) The singular cochain complex with coefficients in M. Definition 2.2. 0 C•(X; X ; M) The relative chain complex with coefficients in M. Definition 1.27. C•(X; X0; M) The relative cochain complex with coefficients in M. Above Proposition 2.12. CMon The category of commutative monoids. Definition 4.25. CohomTh The category of generalised cohomology theories on CW-complexes. Beginning of Section 5.2. CW The category of CW-complexes. Notation 3.22. cCW The category of path-connected CW-complexes. Notation 6.27. Hn(X; M) The n-th singular homology module with coefficients in M. Definition 1.16. Hn(X; M) The n-th singular cohomology module with coefficients in M. Definition 2.6. 0 Hn(X; X ; M) The n-th relative homology module with coefficients in M. Definition 1.27. Hn(X; X0; M) The n-th relative cohomology module with coefficients in M. Definition 2.16. hC The homotopy version of a category C. HomR( · ; · ) The Hom-functor in the category of R-modules. Notation B.3. K(A; n) The (an) Eilenberg-MacLane space of type K(A; n). Notation 6.50. mf , Mf The reduced and unreduced mapping cylinder of a map. Definition 3.44. Map(X; Y ) The set of continuous maps between topological spaces. Notation 1.9. Map•(X; Y ) The set of pointed continuous maps between pointed topological spaces. Notation 6.6. M[S] The linearisation of a set S over a module M.
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