From Obstructions to Invariants: Theory of Obstructions Theory. Adam C. Fletcher

Transcription

1 From Obstructions to Invariants: Theory of Obstructions Theory Adam C. Fletcher c Summer 2010

2 Contents 1 Obstruction Mappings Notes to the Reader Obstructions as Co-chains Obstructions as Co-cycles Properties of Obstructions The Deformation Co-chain Properties of Deformation Co-chains Obstruction Sets Sets of Obstructions to Extension Properties of Obstruction Sets Relative Homotopy The Deformation Co-cycle Application of the Deformation Co-cycle Sets of Obstructions to Homotopy Properties of Homotopy Obstruction Sets Application of Obstructions The Fundamental Homotopy Lemma Classification via Characteristic Element Primary Obstructions Properties of Primary Obstructions End Remarks to the Reader

3 Abstract The generalization of everyday concepts, and applying rigor thereunto, has long intrigued the mathematician. The topic of an obstruction theory arises from the question of deforming continuous maps, and what, exactly, holes are. Intuitively, the fundamental difference between an orange and a doughnut is that a doughnut has a hole, and an orange does not. When shrinking a rubber band around an orange, the rubber band will vanish; whereas the rubber band that passes through the hole of the doughnut, or the one that encircles the doughnut at its widest point, cannot vanish without cutting the doughnut or snapping the rubber band. When we apply the mathematical rigor that transforms oranges, doughnuts, and rubber bands into spheres, tori, and copies of S 1, we raise the question of what the empty space in the torus actually represents. The shrinking of the rubber band represents continuous deformations of circles and spheres. What, then, will obstruct our continuous deformation? The answer to this question forms the foundation of an obstruction theory.

4 Chapter 1 Obstruction Mappings 1.1 Notes to the Reader It should be noticed that there are a number of ways to develop the theory of obstructions to extensions and, thence, homologies. Since obstructions will be built using the tools of cohomology, one realizes that there should be definitions using cellular cohomology, singular cohomology, and simplicial cohomology. Moreover, there could be processes built from Alexander, Čech, or de Rham cohomologies. It can be shown that whenever a collection of cohomology theories are proven to be equivalent, their respective theories of obstruction will be, as well. Within this study, we follow, predominately, the CW-complex approach, as do Hu ([4]) and Whitehead ([9]); whereas Boltyanskii ([1]) and Olum ([7]) engage in an almost identical methodology, but using the simplicial definitions, instead. Again, since the cohomologies are equivalent, so, too, are the obstruction theories. For requisite definitions and theorems not contained in this article, the reader is referred to [2], of which this is a continuation. 1.2 Obstructions as Co-chains Let K be a polyhedron and L be a subpolyhedron thereof. Then a CW-complex structure may be placed on K and inherited by L. Moreover, let Y be a pathconnected, n-simple space (that is, a space for which the fundamental group acts simply on π n (Y )). Let σ be an (n+1)-cell of K. This grants that the boundary of σ is an n-sphere, and is, therefore, contained in the n-skeleton of K. For any g : K n Y, then, the restriction, g σ, of g to σ is the restriction of g to an n-sphere in K n. In 1

5 symbols, σ = S n K n K n. Notice that, by construction, g σ is an element of π n (Y ). Define a map c n+1 (g) : K n+1 K n π n (Y ) by [c n+1 (g)](σ) = [g σ ] for each (n + 1)-cell of K. As an induced map from π n+1 (K) π n (Y ), we have that c n+1 (g) is a (n + 1)-cochain on K. We call this cochain the obstruction of the map g. Example. Consider the Möbius band. Let g be the map defining the complex structure of the Möbius band. Let σ 1 be the boundary circle, and let σ 2 be the interior disc. Notice that, while [c 1 (g)](σ 1 ) = [g σ 1 ] = [g 0] = 0 seems to represent the lack of obstruction, and would, therefore, allow extension over 1-cells; the second obstruction, [c 2 (g)](σ 2 ) = [g σ 2 ] = [g S 1 ] is absorbed by the boundary circle of the Möbius band. This seems to create an obstruction to the attachment of discs to the Möbius band, and explains why the Möbius band is nonorientable. Figure 1.1: The Möbius Band Example. Consider the usual torus in R 3. Let σ 1 be either of the identified edges (the equator or the meridian) of the torus; let σ 2 be the interior disc that is mapped by the characteristic map to the surface of the torus. Moreover, consider σ 3, some solid sphere. Now, consider g as the map defining the complex structure on the torus. Notice that, while [c 1 (g)](σ 1 ) = [g σ 1 ] = [g 0] = 0 2

6 seems to allow extension on 1-cells, and [c 2 (g)](σ 2 ) = [g σ 2 ] = [g aba 1 b 1 ] = [g 0] = 0 under the quotient topology (where a and b are the identified edges), appears to grant orientability, we see an issue with this third obstruction class. Notice that [c 3 (g)](σ 3 ) = [g σ 3 ] = [g S 2 ], will represent the donut hole in the center of the torus and the inner jelly filling. This will create an obstruction to the attachment of solid spheres to the torus, which is known as handling. Figure 1.2: Obstructions in a Torus More rigor will be applied to these examples below. 1.3 Obstructions as Co-cycles In order to apply rules of cohomology to the idea of an obstruction to extensions, we wish to classify the obstruction as either a cocycle or a coboundary, as these are the primary concerns of a cohomology group. Proposition For g : K n Y, the obstruction cochain c n+1 (g) is, in fact, an (n + 1)-cocycle over K. Proof. Let σ be an (n + 2)-cell in K. Then σ K n+1. It suffices to show that the coboundary of the chain is zero; that is, that δc n+1 (g)(σ) = 0. To this end, 3

7 we consider the following chain of homomorphisms: C n+1 ( σ) Z n ( σ) by the long exact cohomology sequence = Z n (( σ) n ) since there are no n-cycles in the (n + 1)-skeleton = H n (( σ) n ) by the quotient group definition of homology h π n (( σ) n ) via the natural abelianization homomorphism (g σ ) πn (Y ), as defined above. If n > 1, we recall that ( σ) n is (n 1)-connected, since m-spheres are nullhomotopic over n-spheres for each m < n. The theorem of Hurewicz then gives h to be an isomorphism. On the other hand, if n = 1, it is well-known that h is surjective, and it can be shown that ker h ker g σ. Thus, the composition (g σ ) h 1 : Z n ( σ) π n (Y ) behaves well. In all cases, then, this homomorphism is well-defined. Since C n 1 ( σ) is a free abelian group, for C n ( σ) = ker im : C n ( σ) C n 1 ( σ). It should be noticed, however, that Z n ( σ) = ker, and so, Z n is a summand of C n. We may then extend the map (g σ ) h 1 to a map d : C n ( σ) π n (Y ). Then, for each (n + 1)-cell, σ, in σ, we have so [c n+1 (g)](σ ) = g σ = (g σ) σ = (g σ ) h 1 ( σ ) = d( σ ), [δc n+1 (g)](σ) = [c n+1 (g)]( σ) = d( ( σ)) = d(0) = 0. Thus, c n+1 (g) has no coboundary, and is an (n + 1)-cocycle in K. 1.4 Properties of Obstructions With the definition of the obstruction cocycle, some properties make themselves known. With an eye forward to determining which topological spaces are the same, and classification theorems, we note these properties here. 4

8 Proposition Let g : K n Y. Then g may be extended over K n+1 if and only if c n+1 (g) = 0. Proof. We notice first that, if σ is an (n + 2)-cell in K, and g : K n+1 Y is an extension of g, we have that c n+1 (g) = c n+1 (g σ) = [(g σ) ( σ)] = [g 0] = 0. Conversely, if c n+1 (g) = 0, it means that g is a coboundary on K. That is, there is a map h : K n+1 Y so that g = δh = h. Then, for any (n + 1)-cells, τ, in K, the image g(τ) is the same as that of the boundary under the higher map, h( τ). Thus, h agrees with g on its boundary, so h is the desired extension. Proposition Let f, g : K n Y be homotopic maps. Then the resulting obstructions are equal. Proof. Let σ be an (n + 1)-cell in K. Then, if f, g are homotopic maps, we have that the restrictions of each map to σ must also be homotopic. The restrictions, then, are in the same class; that is, are the same element, of π n (Y ). By construction, then, c n+1 (f) = c n+1 (g), as desired. Proposition Let φ : K 1 K 2 be a cellular map between polyhedra. Let g 1 : K n 1 Y and g 2 : K n 2 Y so that g 1 = g 2 φ. Then, under the induced cohomological map φ #, we have c n+1 (g 1 ) = φ # c n+1 (g 2 ). Proof. Let σ be an (n + 1)-cell over K 1. Since σ is arbitrary, it will suffice to show that c n+1 (g 1 )(σ) = φ # c n+1 (g 2 )(φσ), where φσ is the direct image of σ under φ. Since φ is a cellular map, we know that φσ is an (n + 1)-cell over K 2. We consider the following diagram, K_1! K_2 g_1 g_2 C_{n+1}(K_1)!^# C_{n+1}(K_2) c^{n+1}(g_1) c^{n+1}(g_2) Y Y "_n(y) "_n(y) Figure 1.3: Obstruction-Preserving Cellular Maps and notice that, since applying cohomological chains induces contravariant maps (hence, the unconventional, yet more precise, terminology in [3]), the top 5

9 brown arrow reverses direction in Figure 1.3. A number of commutativity properties of the diagram give the following system of equalities. c n+1 (g 1 )(σ) = [ g 1 σ ] and, so, c n+1 (g 1 ) = φ # c n+1 (g 2 ), as desired. = [ g 2 φ σ ] = [ g 2 φ( σ) ] = φ # [ g 2 (φσ) ] = φ # c n+1 (g 2 )(φσ), We see from these properties, then, that the obstruction cocycle of a map is a homotopy invariant, and can be used to differentiate between non-homotopic maps. Hence, if the examples from section 1.2 are to be believed, we see that trivial loops on the torus and its equator are non-homotopic, as can be seen by the former vanishing, and the latter not. Moreover, there is no cellular map on 1-cells between the Möbius band and the torus, as c 2 (g) would need to be preserved, and orientability considered. 1.5 The Deformation Co-chain There is, in particular, a very important class of co-chains in the theory of obstructions. Just as the generalization of lifting properties from extensions to homotopies gives a number of important results, so, too, does the generalization of obstruction properties. In order to generalize the theory to obstructions of homotopies, we need a very special co-chain. Let g 0, g 1 : K n Y be homotopic on the subpolyhedron K n 1. Then there is a homotopy h t : K n 1 I Y so that h 0 is the restriction of g 0 to K n 1 and h 1 is the corresponding restriction of g 1. We then define the map Φ : (K I) n Y by g 0 (x) x K n, t = 0 Φ(x, t) = h t (x) x K n 1, 0 < t < 1. g 1 (x) x K n, t = 1 It can be seen, then, that the obstruction c n+1 (Φ) agrees with c n+1 (g 0 ) and with c n+1 (g 1 ) at the appropriate values of t. It is clear, then, that c n+1 (Φ) c n+1 (g 0 ) 0 c n+1 (g 1 ) 1 6

10 is a cochain over K I, again with coefficients in π n (Y ). Since, for each cell σ, there is a homeomorphism σ σ t between K and K I, this gives rise to an isomorphism k between the n-chains of K and the (n + 1)-chains of K I. In particular, there is a unique cochain d n (g 0, g 1 ; h t ) so that kd n (g 0, g 1 ; h t ) = ( 1) n+1 [ c n+1 (Φ) c n+1 (g 0 ) 0 c n+1 (g 1 ) 1 ]. Definition The unique n-cochain d n (g 0, g 1 ; h t ) over K so that kd n (g 0, g 1 ; h t ) = ( 1) n+1 [ c n+1 (Φ) c n+1 (g 0 ) 0 c n+1 (g 1 ) 1 ] is called the deformation cochain. Moreover, if g 0 and g 1 agree on the (n 1)-skeleton, and their homotopy is, say, g 0, then the deformation cochain simplifies to d n (g 0, g 1 ) = c n+1 (g 0 ) c n+1 (g 1 ). This shall be called the difference cochain, and shall be considered in proposition below. It should be noted here, however, that the literature is not consistent. While we follow [4] with our definition, which is more general, some (like [1]) require equality rather than homotopy. Here, the boundary-type homomorphism D # : C n+1 (K I) C n (K) can be applied to the obstruction c n+1 (Φ), granting the same formula as the deformation cochain. This requirement, however, merely simplifies the theory to the study of the difference cochain, and does not lose any rigor. 1.6 Properties of Deformation Co-chains With the definition of the deformation cochain, some properties make themselves known. With an eye forward to determining which topological spaces are the same, and classification theorems, we note some of these properties here. It should be mentioned that deformation and difference co-chains are merely specific types of obstruction co-chains, and so, previous properties may simplify our proofs. Proposition Let g 0, g 1 : K n Y be homotopic, and h t be an appropriate homotopy. Then h t may be extended over K n+1 if and only if d n (g 0, g 1 ; h t ) = 0. Proof. In order for the homotopy h t to extend over K n+1, both base functions (g 0 and g 1 ) must extend. Moreover, the function Φ, by continuity, must extend. These will occur, by proposition 1.4.1, if and only if each of c n+1 (g 0 ) = 0, 7

11 c n+1 (g 1 ) = 0, and c n+1 (Φ) = 0. When all three parts simultaneously vanish, however, we have kd n (g 0, g 1 ; h t ) = ( 1) n+1 (0 0 0) = 0. Since k is an isomorphism, the only element of ker k is the zero element. Thus, the homotopy extends over K n+1 if and only if the deformation cochain vanishes, as desired. Proposition (The Coboundary Formula). Let g 0, g 1 : K n Y be homotopic, and h t be an appropriate homotopy. Then δd n (g 0, g 1 ; h t ) = d n (g 0, g 1 ). Proof. Let σ be an (n + 1)-cell over K I. Then we consider the image of the cell under the deformation cochain kd n (g 0, g 1 ; h t ), and, in turn, under the coboundary map. Since (co-)boundary operators commute, we have δkd n (g 0, g 1 ; h t )(σ) = kd n (g 0, g 1 ; h t )( σ), which, by the definition of our deformation cochain, is equivalent to ( 1) n+1 [ c n+1 (Φ)( σ I) c n+1 (g 0 )( σ) 0 c n+1 (g 1 )( σ) 1 ]. Making use of a reduction formula (cf. [9], p. 68) for the boundary of a cylinder (in particular, of σ I) yields ( 1) n+1 [ c n+1 (Φ)(σ 1 σ 0 (σ I)) c n+1 (g 0 )( σ) 0 c n+1 (g 1 )( σ) 1 ], which, in turn, equals ( 1) n+1 [ c n+1 (Φ)(σ 1) c n+1 (Φ)(σ 0) c n+1 (Φ)( (σ I)) c n+1 (g 0 )( σ) 0 c n+1 (g 1 )( σ) 1 ]. Now, since Φ is defined in terms of g 0 and g 1, we may resubstitute, yielding ( 1) n+1 [ c n+1 (g 1 )(σ) c n+1 (g 0 )(σ) c n+1 (Φ)( (σ I)) c n+1 (g 0 )( σ) 0 c n+1 (g 1 )( σ) 1 ]. On the other hand, since c n+1 (g 0 ), c n+1 (g 1 ), and c n+1 (Φ) are cocycles, their (n + 1)-th obstructions over the n-spherical boundary of a cell vanish, and we have ( 1) n+1 [ c n+1 (g 1 )(σ) c n+1 (g 0 )(σ) ], and so is equal to the difference cochain, kd n (g 0, g 1 ). Since k is an isomorphism, and commutes with δ, we have that as desired. δd n (g 0, g 1 ; h t ) = d n (g 0, g 1 ), 8

12 It seems rather futile, however, for us to continue our foray into the realm of obstruction maps if no such maps truly exist. We have, in the magical way, assumed the object of our study exists, and have shown some properties thereof. We, in a way, assume that, just because we wish it to be so, it is. The next proposition should set our minds at ease. Proposition (The Existence Property). Let g 0 : K n Y. If ξ is an n-cochain over K, then there is a map g 1 : K n Y homotopic to g 0 so that ξ is the deformation cochain between g 0 and g 1. Proof. Let σ be an n-cell over K. Then (σ I) is an oriented n-sphere, and, hence, has an attaching map f σ : (σ I) Y that represents ξ(σ) π n (Y ). We recall that (σ I) = (σ 0) ( σ I) (σ 1), and, since we are not yet confident in the end cap, we set τ = (σ 0) ( σ I) (σ I). We note, then, that τ is contractible, and so any two maps in τ are (null-) homotopic. By the homotopy extension property, we may then set { g 0 (x) x σ, t = 0 f σ (x, t) = h t (x) x σ, t I. We then define the map g 1 : K n Y by { h g 1 (x) = 1 (x) f σ (x, 1) x K n 1. x σ It is our claim that d n (g 0, g 1 ; h t ) = ξ. We notice that the obstructions will cancel with the exception of that of f σ, which is represented by ξ(σ) π n (Y ). Since deformation and difference cochains exist as surely as do any cochains, we would like to know that the collections of cochains behave well. With an eye to the cohomology groups from which obstructions arose, we continue to consider their properties. Proposition (The Identity Property). Let g : K n Y. Then both the deformation cochain and the difference cochain d n (g, g) vanish. Proof. Trivially, we notice that kd n (g, g) = ( 1) n+1 [ c n+1 (g) c n+1 (g) ] = 0, as desired for the difference cochain. As to the deformation cochain, we consider the straight line homotopy. This gives Φ(x, t) = g(x) for all 0 t 1, and 9

13 so the t-coordinate is suppressed. That is to say, the Cartesian product of obstructions is, in fact, scalar multiplication. Thus, kd n (g, g; h t ) = ( 1) n+1 [ c n+1 (Φ) c n+1 (g) 0 c n+1 (g) 1 ] = ( 1) n+1 [ c n+1 (g) 0 c n+1 (g) ] as desired. = 0, Proposition (The Addition Property). Let g 0, g 1, g 2 : K n Y be homotopic, and h t, k t be appropriate homotopies. Then, for { l t = h 2t 0 t 1 2 k 2t t 1, we have d n (g 0, g 1 ; h t ) + d n (g 1, g 2, k t ) = d n (g 0, g 2 ; l t ). Proof. Let σ be an n-cell in K. We suppress the powers of 1 in the definition of the deformation cochain, and see that d n (g 0, g 2 ; l t ) = c n+1 (Φ)(σ I) c n+1 (g 0 )(σ) 0 c n+1 (g 2 )(σ) 1. We notice, however, that our homotopy is split into two pieces. Therefore, we consider I 1 = [ 0, 2] 1 and I2 = [ 1 2, 1] as the partition of the unit interval, and σ 1 σ 2 as a partition of σ, where σ 1 is covered in the time taken to traverse I 1, and similarly for σ 2. Then, Φ = Φ 1 + Φ 2, defined similarly, and c n+1 (Φ)(σ I) = c n+1 (Φ 1 )(σ 1 I 1 ) + c n+1 (Φ 2 )(σ 2 I 2 ). Moreover, we see that, at t = 1/2, g 1 is defined on both σ 1 and σ 2, but with opposite orientations. Thus, c n+1 (g 1 )(σ 1 ) = c n+1 (g 1 )(σ 2 ). Then we have d n (g 0, g 2 ; l t ) = c n+1 (Φ)(σ I) c n+1 (g 0 )(σ) 0 c n+1 (g 2 )(σ) 1 = c n+1 (Φ 1 )(σ 1 I 1 ) + c n+1 (Φ 2 )(σ 2 I 2 ) c n+1 (g 0 )(σ 1 ) 0 c n+1 (g 2 )(σ 2 ) 1 = c n+1 (Φ 1 )(σ 1 I 1 ) + c n+1 (Φ 2 )(σ 2 I 2 ) c n+1 (g 0 )(σ 1 ) 0 c n+1 (g 2 )(σ 2 ) 1 c n+1 (g 1 )(σ 1 ) c n+1 (g 1 )(σ 1 ) 1 2 = c n+1 (Φ 1 )(σ 1 I 1 ) + c n+1 (Φ 2 )(σ 2 I 2 ) c n+1 (g 0 )(σ 1 ) 0 c n+1 (g 2 )(σ 2 ) 1 c n+1 (g 1 )(σ 1 ) 1 2 c n+1 (g 1 )(σ 2 )

14 After some rearranging terms, we may consider h t on σ 1 and I 1, while k t applies over σ 2 and I 2. Thus, we have the above equivalent to as desired. d n (g 0, g 1 ; h t ) + d n (g 1, g 2 ; k t ), Proposition (The Inverse Property). Let g 0, g 1 : K n Y be homotopic, and h t be an appropriate homotopy. Then, for h t = h 1 t, we have d n (g 0, g 1 ; h t ) = d n (g 1, g 0, h t ). Proof. This follows directly from proposition 1.6.5, proposition 1.6.4, and standard subtraction. As deformation cochains inherit associativity from general cochains, we see that the collection of deformation cochains now forms an algebraic group. This, however, should not seem surprising, in light of proposition 1.6.3, since this group is merely that of the cochains on K. As a direct result of proposition 1.4.3, we may state the following proposition without proof. Proposition Let φ : K 1 K 2 be a cellular map between polyhedra. Let f 1 and g 1 : K n 1 Y, with f 2 and g 2 : K n 2 Y so that f 1 = f 2 φ and g 1 = g 2 φ. Further, let h 1,t and h 2,t be associated homotopies. Then, under the induced cohomological map φ #, we have d n (f 1, g 1 ; h 1,t ) = φ # d n (f 2, g 2 ; h 2,t ). As we have seen before, this results in the deformation and difference cochains being topological invariants over their respective polyhedra. 11

15 Chapter 2 Obstruction Sets As is important to do in any mathematical theory, we have studied the objects of obstruction theory, the obstruction cocycle itself; we have studied the morphisms between such objects, in the guise of the deformation and difference cochains, and the cellular extensions. Now, we study the collections of these objects. 2.1 Sets of Obstructions to Extension As above, we consider the polyhedron K and the path-connected, n-simple space Y. For any g : K n Y we have the obstruction cocyle c n+1 (g), which may well be zero. As when we saw this element as a representative of a class in π n (Y ), the obstruction also determines an element (equivalence class) of the cohomology group H n+1 (K). These equivalence classes, denoted γ n+1 (g), of these obstructions, then, form a subset of the cohomology group. Definition The set containing the equivalence classes of the obstructions γ n+1 (g) of a map g : K n Y is called the obstruction set of g, and will be denoted O n+1 (g). Recall that a map g is said to be n-extensible over K if g can be extended over K n for n 0. Of course, since Y is path-connected, any g : K Y is 1-extensible. Moreover, the supremum of the set of n for which g is n-extensible is called the extension index of g over K. Example. We note that, if g is not n-extensible, then it would be silly to discuss an obstruction to extending g to the (n + 1)-skeleton of the polyhedron. In this case, we take O n+1 (g) =, or say that the obstruction set is vacuous ([4]) or void ([7]). Example. The next smallest obstruction set would be of a map onto a contractible space. Since a polyhedron will be contractible if and only if π n (K) = 0 for all n, the only allowable maps are null-homotopic ones. Thus, the obstruction set is O n+1 (g) = 0. This can be called a null obstruction class. 12

16 As a matter of interest, it should be noted that, although polyhedra follow the rule that null homotopy groups are necessary and sufficient for contractibility, there are topological spaces that are not contractible, but do have null homotopy groups. Zeeman s cone, as discussed in [3], is such a space with no CW-complex structure to be placed upon it. Figure 2.1: Zeeman s Cone 2.2 Properties of Obstruction Sets With the definition of the obstruction set of a map, some properties make themselves known. With an eye forward to determining which topological spaces are the same, and classification theorems, we note some of these properties here. Throughout, as before, we let K be a polyhedron upon which a CW-complex structure has been defined, and Y be a path-connected space. Moreover, the maps f, g : K n Y shall be our standard maps. Proposition Let f and g be homotopic. Then O n+1 (f) = O n+1 (g). Proof. We recall, by proposition 1.4.2, that, as homotopic maps, c n+1 (f) = c n+1 (g), and, therefore, define the same equivalence classes in H n+1 (K). Thus, as desired. O n+1 (f) = O n+1 (g), Proposition If φ is a cellular map from K 1 to K 2, and f and g are defined on the appropriate polyhedra with g = φf, then φ : O n+1 (f) O n+1 (g). Proof. As we see in proposition 1.4.3, we have that c n+1 (g) = φ # c n+1 (f). That is to say, that for each obstruction of f, there is a corresponding obstruction of g, which is the image under φ #. It is clear, then, that φ # induces a map between the cohomology groups, and, in particular, the obstruction sets, as desired. 13

17 Proposition The map g is (n + 1)-extensible over K if and only if its obstruction set is null; that is, if and only if O n+1 (g) = {0}. Proof. We recall, from proposition 1.4.1, that g is (n + 1)-extensible if and only if c n+1 (g) = 0. Moreover, by a slight generalization of proposition 1.6.3, called Eilenberg s extension theorem (cf. [4], VI.5), we have that the nullity of c n+1 (g) and the equivalence class, γ n+1 (g) are equivalent. Since this null class is the only such class, 0 = O n+1 (g) as desired. Proposition The obstruction set O n+1 (g) in nonempty if and only if g is n-extensible over K. Proof. Notice that, if g is n-extensible over K, then either there is an extension to K n+1 or there is not. If there is, there is no obstruction to extension, and c n+1 (g) = 0. Therefore, γ n+1 (g) = 0, and 0 O n+1. If there is no extension to K n+1, there is some obstruction to the extension, c n+1 (g), and the class of this obstruction is an element of O n+1 (g). Therefore, in either case, O n+1 (g). On the other hand, if g is not n-extensible, we have, by our first example in section 2.1, the obstruction set will be void. In either case, then, vacuity of the (n + 1)-dimensional obstruction set is tied directly to the n-extensibility of the requisite map. We see from these properties, then, that the collection of obstruction sets of a map form a homotopy invariant, and can be used to differentiate between non-homotopic mappings and their domain spaces. 2.3 Relative Homotopy Throughout, we have been concerned with only one major obstructions: those to extending a map on the n-skeleton of a polyhedron to a map on its (n + 1)- skeleton. We have, however, seen that obstructions to extensions form a number of homotopy invariants. The next step in the study of obstructions, then, should be of determining obstructions to finding a homotopy to given maps. It is with this question that we concern ourselves in this section. In order to study the idea of obstructions to homotopies, we will need to recall a few terms, and so make the following Definition Two maps f : K Y and g : K Y on a polyhedron that agree on a subpolyhedron L are said to be homotopic relative to L when there is a homotopy h t : K Y so that h 0 = f, h 1 = g, and h t L = f L for each t (0, 1). We denote this f L g. Definition Two maps f : K Y and g : K Y so that f K n L g K n are said to be n-homotopic, and shall be denoted f n L g. 14

18 One should notice, of course, that since, throughout, Y has been path-connected; so, if f and g agree on some polyhedron L, they are 0-homotopic relative to L. Definition The supremum of the set of n N such that f n L g is called the homotopy index. It should be fairly straightforward to note that If f L f and g L g, then the homotopy indices of the pair (f, g) and of the pair(f, g ) are the same. Hence, homotopy index is a topological invariant. 2.4 The Deformation Co-cycle We will now make use of the deformation cochain, as defined above, as a descriptor of an obstruction to homotopy. To this end, we consider any two maps f, g : K Y which are (n 1)-homotopic, relative to the subpolyhedron, L. As the maps are (n 1)-homotopic, there is a homotopy h t : K n 1 Y with the properties that h 0 = f K n 1, h 1 = g K n 1, and h t L = f L for each 0 < t < 1. Since f, g are defined on all of K, they are, in particular, defined over K n, and so, we have a deformation cochain, d n (f, g; h t ). It is this cochain which we claim will serve as the obstruction to homotopy between the pair (f, g). Proposition The deformation cochain, d n (f, g; h t ), is, in fact, a cocycle. Proof. Since f, g are defined on K n+1, there is no obstruction to their extensions from the n- to the (n + 1)-skeleton. Thus, c n+1 (f) = 0 and c n+1 (g) = 0. By proposition 1.6.2, we have that and, so, d n (f, g; h t ) is a cocycle. δd n (f, g; h t ) = c n+1 (f) c n+1 (g) = 0 0 = 0, Proposition The homotopy h t defined above has an extension to the n-skeleton if and only if d n (f, g; h t ) = 0. Proof. As defined in section 1.5, we have a Φ : K I n Y that results in an obstruction c n+1 (Φ). We have, for the chain isomorphism, k, that kd n (f, g; h t ) = ( 1) n+1 [c n+1 (Φ) c n+1 (f) 0 c n+1 (g) 1], but, since f and g extend, their obstructions are null. Thus, kd n (f, g; h t ) = ( 1) n+1 c n+1 (Φ). 15

19 Thus, Φ (and, thereby, h t ) extends if and only if c n+1 (Φ) = 0, or, since k is an isomorphism, and, hence, has only the zero element in its kernel, if and only if the deformation cocycle, d n (f, g; h t ) is null. This cocycle, then, results in a representative obstruction cohomology class, and, as before, δ n (f, g; h t ) H n (K, L; π n (Y )), Theorem (Eilenberg s Homotopy Theorem). The obstruction cohomology class δ n (f, g; h t ) is null if and only if there exists a homotopy h t : K n Y with the properties that h 0 = f K n, h 1 = g K n, h t K n 2 = h t K n Application of the Deformation Co-cycle Just as the sets of obstructions to the extension of maps was defined, we would like to define sets of obstructions to homotopies of pairs of maps. To this end, we will use the deformation cocycle to define a number of sets whose interactions will define the desired obstruction sets. We first fix a map f : K Y. This map is the base map for the homotopy pair. An allowable, (n 1)-homotopic pair is needed to proceed further. To this end, we define the set Ω = {g : K n 1 Y } and one of its subsets, W = {g Ω g L f}. The set W contains all maps that are, relative to the subpolyhedron L, (n 1)-homotopic to f. We notice that, trivially, f is (n 1)-homotopic to itself, relative to L, and so would form an allowable pair. For notational convenience, then, we define w 0 = f K n 1 W. We then define the set R n (K, L; f) to be π 1 (W, w 0 ). This is a collection of the equivalence classes of homotopies of f onto itself, restricted to the closed (n 1)- skeleton. These homotopic loops allow f to deform, and, yet, like a vibrating rubber band, snap back to itself eventually. These elements (homotopies), then, satisfy the requirements above for the construction of a deformation cochain, d n (f, f; h t ); and, thence, a deformation obstruction class, δ n (f, f; h t ). It should be noticed, however, that δ n (f, f; h t ) depends only on the homotopic route or path through which f vibrates. We temporarily denote this path as α R n (K, L; f). 16

20 Proposition There is a correspondence α ξ n (α) = δ n (f, f; h t ) that gives rise to a homomorphism ξ n : R n (K, L; f) H n (K, L; π n (Y )). Proof. Since ξ n is defined in terms of δ n, it is a well-defined map; and the Addition Property (1.6.5) gives the preservation of operation required of a homomorphism. It should be noted, by the way, that, although our case seems convoluted and confusing, it seems to be the simplest case of finding obstructions to homotopies. Changing the range/codomain to a space that is not n-simple creates a crossed homomorphism ([4]), and removing the relative homotopy condition leads to the study of obstructions to free homotopy, which also introduces crossed homomorphisms ([7]). Returning to our simplified case, we define the set J f n(k, L; π n (Y )) as the image of R n (K, L; f) in H n (K, L; π n (Y )) under the homomorphism ξ n. Moreover, we notice that J f n is a subgroup of H n, and so denote by Q f n the quotient group Q f n(k, L; π n (Y )) = Hn (K, L; π n (Y )) J f n(k, L; π n (Y )). Proposition This group J f n depends only on the (n 1)-th homotopy class (relative to L) of f. Proof. Let g : K Y be (n 1)-homotopic to f, relative to L (that is, g W ). Then it suffices to show that J f n = J g n. In fact, one inclusion will grant the other inclusion by symmetry, mutatis mutandi. Thus, it suffices for us to show that J f n J g n. Since g is (n 1)-homotopic to f, there is a homotopy k t : K n 1 Y with k 0 = f K n 1, k 1 = g K n 1, and k t L = f L for each t in the unit interval. For any α R n (K, L; f), we have δ n (f, f; h t ) = ξ n (α) J f n, and so ξ n (α) is associated with the representative cocycle element d n (f, f; h t ) of the obstruction class. We define, then, a composite homotopy, l t : K n 1 Y as follows: k 1 3t (x) x K n 1, t [ ] 0, 1 3 l t (x) = h 3t 1 (x) x K n 1, t [ 1 3, ] 2 3 k 3t 2 (x) x K n 1, t [ 2 3, 1]. It should be noticed, then, that l 0 = g K n 1 ; that l 1 = g K n 1, as well; and that l t L = f L for each t (0, 1). On the other hand, f and g are to agree on the subpolyhedron L, so l t L = g L for each t [0, 1]. Thus, l t represents a 17

21 homotopy of g onto itself, or some β R n (K, L; g). On the other hand, we have that, by the Addition and Inverse Properties (1.6.5 & 1.6.6) and described by the cases for l t, we have that d n (g, g; l t ) = d n (g, f; k t ) + d n (f, f; h t ) + d n (f, g; k t ) = d n (f, g; k t ) + d n (f, f; h t ) + d n (f, g; k t ) = d n (f, f; h t ). Hence, the images of the homotopy paths are equal: ξ n (α) = d n (f, f; h t ) = d n (g, g; l t ) = ξ n (β), the last of which is an element of J g n. So arbitrary elements of J f n are contained in J g n, and inclusion (and, hence, equality) holds. 2.6 Sets of Obstructions to Homotopy When considering the obstructions to extension of maps, we defined the obstruction set O n+1 (f) as the collection of obstruction classes γ n+1 (f) of equivalence classes of obstruction cocycles c n+1 (f). Here, we wish to build a similar set, only consisting of obstructions to homotopies between pairs of maps, (f, g). We note that, if (f, g) is not an (n 1)-homotopic pair, it would be silly to discuss an obstruction to homotoping f to g. In this case, we take O n (f, g) =, or say that the obstruction set is vacuous ([4]) or void ([7]). If, on the other hand, (f, g) is an (n 1)-homotopic pair, there is a homotopy h t : K n 1 Y for t [0, 1] satisfying the properties outlined above. This gives rise to the deformation cocycle, d n (f, g; h t ), and, in turn, its obstruction element δ n (f, g; h t ). We define the set of obstructions to homotopy of the pair (f, g) to be the collection of these equivalence classes. The n-dimensional set is denoted O n (f, g). 2.7 Properties of Homotopy Obstruction Sets With the definition of the obstruction set of a map, some properties make themselves known. With an eye forward, as always, to determining which topological spaces are the same, and classification theorems, we note some of these properties here. All notation is as above. Proposition Let f L f and g L g. Then O n (f, g) = O n (f, g ). Proof. Since pairs of homotopic maps have the same homotopy index, this statement is straightforward. 18

22 Proposition If φ is a cellular map from K 1 to K 2, and (f, g) and (f, g ) are defined on the appropriate polyhedra with f = φf and g = φg, then φ : O n (f, g) O n (f, g ). Proof. As we see in proposition 1.6.7, we have that d n (f, g ; h t) = φ # d n (f, g; h t ). That is to say, that for each obstruction of the pair (f, g), there is a corresponding obstruction of the pair (f, g ), which is the image under φ #. It is clear, then, that φ # induces a map between the cohomology groups, and, in particular, the obstruction sets, as desired. We see from these properties, then, that the collection of homotopy obstruction sets of a map form a topological invariant, and can be used to differentiate between pairs of non-homotopic mappings and their domain spaces. Proposition The maps f and g have the property f n 1 L g if and only if O n (f, g) is a coset of J f n(k, L; π n (Y )) H n (K, L; π n (Y )). Proof. Let f and g be (n 1)-homotopic maps, relative to the subpolyhedron L. As such, we have homotopies h t and k t so that δ n (f, g; h t ) and δ n (f, g; k t ) are obstruction elements of the pair (f, g), and we may form the composite homotopy { h 2t (x) t [ ] 0, 1 l t (x) = 2 k 2 2t (x) t [ 1 2, 1] to create a loop of f back to itself. This loop, α, is in R n (K, L; f), as defined in section 2.5, and its image can be represented as the difference or as ξ n (α) = δ n (f, g; h t ) δ n (f, g; k t ), ξ n (α) + δ n (f, g; k t ) = δ n (f, g; h t ). As ξ n (α) J f n, and δ n (f, g; h t ) O n (f, g), we may write, setwise, that O n (f, g) δ n (f, g; k t ) + J f n(k, L; π n (Y )), so that O n (f, g) is contained a coset of J f n. Conversely, if we take a representative coset of J f n, we consider an element α R n (K, L; f), we have a homotopy m t and an image ξ n (α) = δ n (f, f; m t ). Needing to insert the map g into the picture, we recall that, as (n 1)-homotopic maps, there is a homotopy j t satisfying the previously stated conditions. Then, we define the composite homotopy { j 2t (x) t [ ] 0, 1 p t (x) = 2 m 1 2t (x) t [ 1 2, 1], 19

23 which homotops f onto g. We have, by the Addition Property, though, that δ n (f, g; p t ) = δ n (f, f; m t ) + δ n (f, g; j t ) = ξ n (α) + δ n (f, g; j t ). Thus, for each element of a coset (the right-hand side), we have constructed an obstruction elements; and so, cosets of J f n are contained in O n (f, g). By the process of double inclusion, then, we have shown that O n (f, g) is a coset of J f n(k, L; π n (Y )), as desired. To show sufficiency of this condition, however, we notice that if there is no (n 1)-homotopy between f and g, then the n-dimensional obstruction class is empty, and so, is not a (proper) coset of J f n. This last proposition now gives some algebraic hint to calculating the obstruction groups between homotopies. As the reader may have noticed, the lack of examples throughout this article lends weight to the claim that calculating obstructions is deceptively difficult. In the remaining chapter, we plan to pave road to making calculations of obstructions slightly easier. 20

24 Chapter 3 Application of Obstructions 3.1 The Fundamental Homotopy Lemma In order to calculate obstructions, we first need to know whether such an endeavor is worthwhile. If a set of homotopy obstructions is empty, then we move on without pausing. If the set contains a unique element (most likely, the zero element), we rejoice, and, again, move on. It is with these easier cases that we must deal first. To this end, we prove an important Theorem (Fundamental Homotopy Lemma). Let f, g : K Y be maps from a polyhedron K into a path-connected, n-simple space Y. Let L be a subpolyhedron of K. Then f n L g if and only if O n (f, g) = J f n(k, L; π n (Y )). Proof. Let f and g be n-homotopic maps relative to L. Then there exists a homotopy h t : K n Y so that h 0 = f K n, h 1 = g K n, and h t L = f L for each t [0, 1], and, to satisfy Eilenberg s theorem, we let h t = h t K n 1. We therefore apply proposition 2.4.3, and we have that δ n (f, g; h t ) = 0. This gives that the n-th obstruction class contains the identity element, and so is the unique coset of J f n(k, L; π n (Y )). Thus, O n (f, g) = J f n(k, L; π n (Y )). Conversely, if O n (f, g) is the lone coset of J f n, it must contain the identity (zero) element of the cohomology group, so there is a homotopy h t so that h 0 = f K n 1, h 1 = g K n 1, and h t L = f L with δ n (f, f; h t ) = 0. Moreover, again by Eilenberg, 2.4.3, there is an extended homotopy, h t so that h 0 = f K n, h 1 = g K n, and h t K n 2 = h t K n 2. 21

25 By definition, then, f and g are n-homotopic, relative to L. We recall that the quotient group Q f n is defined by Q f n(k, L; π n (Y )) = Hn (K, L; π n (Y )) J f n(k, L; π n (Y )), and notice that the cardinality of Q f n is the number of cosets that exist of J f n. It follows immediately, then, from proposition 2.7.3, that Proposition If f, g : K Y agree on the subpolyhedron L K and are (n 1)-homotopic relative to L, then Q f n(k, L; π n (Y )) = 1. We define, then, this unique coset as the characteristic element of the pair (f, g), and denote it by χ n (f, g). Moreover, the Fundamental Homotopy Lemma gives that Proposition χ n (f, g) = 0 if and only if f n L g. Moreover, since the homotopy obstruction set is a topological invariant, so too is the characteristic element. 3.2 Classification via Characteristic Element As with any theory in mathematics, the theory of obstructions to extensions and homotopies also looks ahead to the desire to differentiate between maps and their domain spaces that are the same and those that are different, in some sense. At this point, we have the topological technology to study this problem, and attempt to classify the maps and spaces by their obstructions. To this end, we have already defined the characteristic element of a set of obstructions to homotopy of a given map. We must expand upon this element, and consider its properties. First, recall, of course, that K is a polyhedron, L is a subpolyhedron thereof, and Y is a path-connected, n-simple space. We now fix a map µ : L Y. Define the set W = {f : K Y µ = f L} to be the set of extensions of µ over the space K. Then we may divide W into its disjoint equivalence classes under (n 1)-homotopy relative to L, and let θ be one of these classes. Moreover, a portion of the elements in θ are also n-homotopic to each other, and so we further divide θ into its disjoint equivalence classes under n-homotopy relative to L. As above, this division into n-homotopy classes forms the subgroup and therefore, the quotient group J θ n(k, L; π n (Y )) H n (K, L; π n (Y )), Q θ n(k, L; π n (Y )) = Hn (K, L; π n (Y )) J θ n(k, L; π n (Y )). 22

26 Let f be an arbitrary element of θ, and as above, we note that, for each g θ, there is a well-defined χ n (f, g) Q θ n. We shall then say Definition The element α Q θ n is f-admissable if there is a map g θ so that χ n (f, g) = α. These elements, α, form a subset A f n Q θ n, called the f-admissible set. It should then be clear that A f n = χ n (f, g) + A g n for any f, g θ. Moreover, this gives that χ n (f, g) = 0 if and only if f n L g, and so there is an injective correspondence between the number of n-homotopy classes in θ and the elements of f-admissible maps in Q θ n. 3.3 Primary Obstructions With all of the notation that we have defined throughout the study of obstructions, one would think that relationships would be found between them. As always, eyeing the classification theorems, we move closer to this now. We should note at this point, by the way, that [4] uses the what follows as a stepping stone to classification. A similar process is used in [7], however, as an illustrative example, citing the fact that what follows applies not only to Euclidean n-space, but projective n-space and lens spaces, as well. We, as before, require K to be a polyhedron and L to be one of its subpolyhedra, and that Y be n-simple. However, instead of requiring path-connectivity (i.e., 0-connectivity), we require Y to be (n 1)-connected. This is equivalent to the requirement that π k (Y ) = 0 for each k N so that k [1, n). This requirement gives, for any f : L Y, that f can be extended over the n- skeleton of K, and hence, has as its first nontrivial obstruction the set O n+1 (f). By a process similar to that of defining χ n (f, g) above, we find a characteristic element X n (f) representing the function in a cohomology class of L. Therefore we have that Proposition The primary obstruction of extending f over K, ω n+1 (f) is the unique element δ X n (f) O n+1 (f) H n+1 (K, L; π n (Y )), where δ is the coboundary operator on cohomology. Proposition The primary obstruction of homotopy for the pair (f, g) relative to L, ω n (f, g), is the unique element of O n (f, g) so that X n (f) X n (g) = j ω n (f, g), where j is the induced inclusion map on cohomology. This gives that, for each f : K Y, our set J f n is null, making the quotient group equal to the cohomology group of the same dimension. 23

27 3.4 Properties of Primary Obstructions From the uniqueness of primary obstructions to both extensions and homotopies, a number of fairly clear properties present themselves. We list these here, without proof. Proposition For f : L Y, the following are equivalent: 1. f is (n + 1)-extensible over K, 2. ω n+1 (f) = 0; that is, there is no obstruction to extension, and 3. X n (f) is extensible over K. Proposition (Corollary to the Fundamental Homotopy Lemma). Two maps, f and g : K Y agree on L K, and have f n homotopic to g relative to L if and only if ω n (f, g) = 0. Proposition For f, g : K Y, the following are equivalent: 1. f and g are n-homotopic over K, 2. ω n (f, g) = 0; that is, there is no obstruction to homotopy, and 3. X n (f) = X n (g). Moreover, since ω and X are topological invariants, and because of the relationships between Q f n and A f n listed in the previous section, it can be seen that these sets are equivalent to cohomology groups for Y an (n 1)-connected and n-simple space. 3.5 End Remarks to the Reader It seems that, in the realm of well-behaved and nice spaces, the idea of obstruction theory, which was to be a way of defining holes, is tied directly to the idea of a hole being cohomologous/homologous/homotopic/homeomorphic to some k-dimensional sphere. On one hand, the casual reader could be disappointed, as nothing new has seemingly appeared. On the other hand, the reader should be excited that the rigourous algebraic and categorical approach to the study of extension and homotopy relates quite well to the more intuitive, rubber-sheet idea of such things. It should be asked, however, whether this still the case in not-so-well-behaved spaces. The second of Olum s illustrative examples ([7]), discusses obstructions over fibre bundles. In order to explore this example, the author will need to acquaint himself with the basics of fibre bundles, and so, keeping the motivation of obstructions in mind, proceeds thence. Thus endeth the second installment. The reader is invited to continue the journey at 24

28 Bibliography [1] Boltyanskii, V.G. Basic Concepts of Homology and Obstruction Theory. Russ. Math. Surv. 21 (1966), [2] Fletcher, Adam C. An Introduction to the Theory of Obstructions: Notes from the Obstruction Theory Seminar at West Virginia University. Online (2009), [3] Hilton, P.J. and S. Wylie. Homology Theory: An Introduction to Algebraic Topology. Cambridge University Press, London, [4] Hu, Sze-Tsen. Homotopy Theory. Academic Press, New York, [5] Liao, S.D. On the Theory of Obstructions of Fiber Bundles. Ann. Math., II. 60 (July 1954), [6] Mosher, Robert and Martin Tangora. Cohomology Operations and Applications in Homotopy Theory. Harper & Row Publishers, New York, New York, [7] Olum, Paul. Obstructions to Extensions and Homotopies. Ann. Math., II. 52 (July 1950), [8] Steenrod, Norman. The Topology of Fibre Bundles. Princeton University Press, Princeton, New Jersey, [9] Whitehead, George W. Homotopy Theory. MIT Press, Cambridge, Massachusetts,