arXiv:0704.0342v1 [math.AT] 3 Apr 2007 C ro ihrsae rs aual npyis n ognrlz h co the ( Fr¨olich triple generalize a of is do notion [7], A Fr¨olicher Kriegl and the and brielfy physics, manifolds. survey in smooth to naturally is section arise Fr¨olicher this spaces of purpose The Preliminaries 1 FCIP with Cofibrations cofibrations, quence. Smooth retracts, formation Words Key (2000) Classification Subject obain nteCtgr fFr¨olicher Spaces: of Category the in Cofibrations X F • • ⊆ ie iet naaoosrsl htacoe Fr¨olicher su closed deformati a neighborhood that d result smooth analogous neighborhood of an to smooth notion rise The to gives un cofibrations flattened retracts. smooth smooth using tion enable constructed, relate to easily later definition more We classical be the to of extensions analog the ening n nyi h inclusion the if only and Fr¨olicher space Φ Puppe right the construct we application an As cofibrations. X X C R X obain r endi h aeoyo Fr¨olicher spaces of category the in defined are Cofibrations , C ◦ ai iaca taeit Py t,Cp on ot Afr South Town, Cape Ltd, (Pty) Strategists Financial Cadiz F := F¨ lce pcs ltee ntitras mohneighborhood Smooth intervals, unit :Fr¨olicher Flattened spaces, X X { eateto ahmtc n ple Mathematics Applied and Mathematics of Department ⊆ = f : R { X f X ◦ X → afil 02 eulco ot Africa South of Republic 0002, Hatfield uhthat such c | sasot egbroddfrainrtatof retract deformation neighborhood smooth a is Fr¨olicher space mi:[email protected] Email: R f mi:[email protected] Email: | F ∈ f 55P05. : arc ug Ntumba Pungu Patrice ◦ nvriyo Pretoria of University i X c : c , ∈ rt Dugmore Brett A ֒ C X, atI Part Abstract C ∈ → ∞ C ( rsot pc siiilycle by called initially as space smooth or , X X 1 R X ⊆ } , o all for ) F oe rmacransbls of subclass certain a from comes X C ossigo set a of consisting ) ∞ ( c R C ∈ ) X } = bspace F X tintervals. it sequence. nretract on homotopy X yweak- by A eforma- n subsets and , ica up se- Puppe , fthe of X rspaces. er cp of ncept if de- ∞ Γ X := c : R X f c C (R)for all f X = X • F { → | ◦ ∈ ∈F } C Fr¨olicher and Kriegl [7], and Kriegl and Michor [10] are our main reference for Fr¨olicher spaces. The following terminology will be used in the paper: Given a Fr¨olicher space (X, X , X ), the pair ( X , X ) is called a smooth structure; the C F C F elements of X and X are called smooth curves and smooth functions respec- C F tively. The topology assumed for a Fr¨olicher space (X, X , X ) throughout the C F paper is the initial topology induced by the set X of functions. When there TF F is no fear of confusion, a Fr¨olicher space (X, X , X ) will simply be denoted X. The most natural Fr¨olicher spaces are the finiteC dimensionalF smooth manifolds, where if X is such a smooth manifold, then X and X consist of all smooth curves R X and smooth functions X CR. EuclideanF finite dimensional smooth manifolds→ Rn, when viewed as Fr¨olicher→ spaces, are called Euclidean Fr¨olicher spaces. In the sequel, by Rn, n N, we mean the Fr¨olicher space Rn, equipped with its usual smooth manifold∈ structure. A Fr¨olicher space X is called Hausdorff if and only if the smooth real-valued functions on X are point-separating, i.e. if and only if is Hausdorff. TF A Fr¨olicher structure ( X , X ) on a set X is said to be generated by a set X RC F F0 R (resp. C0 X ) if X = ΓF0 and X = ΦΓF0 (resp. X = ΦC0 ⊆ ⊆ C F X F and X = ΓΦC0 ). Note that different sets F0 R on the same set X may giveC rise to a same smooth structure on X. A set⊆ mapping ϕ : X Y between Fr¨olicher spaces is called a map of Fr¨olicher spaces or just a smoot→h map if for each f Y , the pull back f ϕ X . This is equivalent to saying that for each ∈F ◦ ∈F ∞ c X , ϕ c Y . For Fr¨olicher spaces X and Y , C (X, Y ) will denote the collection∈ C of◦ all∈ theC smooth maps X Y . The resulting category of Fr¨olicher spaces and smooth maps is denoted by→FRL. Some useful facts regarding Fr¨olicher spaces can be gathered in the following Theorem 1.1 The category FRL is complete (i.e. arbitrary limits exist ), co- complete (i.e. arbitrary colimits exist), and Cartesian closed.
Given a collection of Fr¨olicher spaces Xi i I , let X = Xi be the set { } ∈ i∈I product of the sets Xi i∈I and πi : X Xi, i I, denote theQ projection map { } → ∈ (xi)i I xi. The initial structure on X is generated by the set ∈ 7→
F = f πi : f Xi . 0 { ◦ ∈F } i[∈I
The ensuing Fr¨olicher space (X, ΓF0, ϕΓF0) is called the product space of the family Xi i I . Clearly, { } ∈ R ΓF = c : X if c(t) = (ci(t))i I , then ci Xi for every i I . 0 { → | ∈ ∈C ∈ }
Now, let Xi be the disjoint union of sets Xi i I , and ιXi : Xi Xi i∈I { } ∈ → i∈I the inclusionU map. Place the smooth final structure on i∈I Xi correspondingU to the family ιXi i∈I . The resulting Fr¨olicher space is calledU the coproduct of { } Xi i I , and denoted Xi, and { } ∈ i∈I ` R i = f : Xi for each i I, f Xi Xi F‘i∈I X { → | ∈ | ∈F } ai∈I
2 is the collection of smooth functions for the coproduct.
Corollary 1.1 Let X, Y , and Z be Fr¨olicher spaces. Then the following canon- ical mappings are smooth. ev: C∞(X, Y ) X Y , (f, x) f(x) • × → 7→ ins:X C∞(Y,X Y ), x (y ins(x)(y) = (x, y)) • → × 7→ 7→ comp:C∞(Y,Z) C∞(X, Y ) C∞(X,Z), (g,f) g f • × → 7→ ◦ f : C∞(X, Y ) C∞(X,Z), f (g)= f g, where f C∞(Y,Z) • ∗ → ∗ ◦ ∈ g : C∞(Z, Y ) C∞(X, Y ), g (f)= f g, where g C∞(X,Z). • ∗ → ∗ ◦ ∈ Given Fr¨olicher spaces X, Y , and Z; in view of the cartesian closedness of the category FRL, the exponential law
C∞(X Y,Z) = C∞(X, C∞(Y,Z)) × ∼ ∞ holds. Because X = C (X, R), it follows by cartesian closedness of FRL that F the collection X can be made into a Fr¨olicher space on its own right. Finally weF would like to show how to construct smooth braking functions, following Hirsch [8]. Smooth braking functions are tools that are behind most results in this paper. In [11], it is shown that the function ϕ : R R given by → 0 if u 0 ϕ(u)= −1 ≤ e u if u> 0 is smooth. Substituting x2 for u in the above function, one sees that the function ψ : R R, given by → 0 if x 0 ψ(x)= −1 ≤ e x2 if u> 0 is smooth. Now, let us construct a smooth function α : R R with the following properties. Let 0 a 0 < α(t) < 1 for a α is strictly increasing for a α(t)=1 for t b. • ≥ 3 Define α : R [0, 1] by → t γ(x)dx α(t)= a , R b a γ(x)dx R where γ(x)= ψ(x a)ψ(b x). − − 1 In the sequel, the notation αǫ, 0 <ǫ< 2 , will refer to a smooth braking function with the following properties αǫ(t)=0 for t ǫ, • ≤ 0 < αǫ(t) < 1 for ǫ α strictly increasing for ǫ αǫ(t)=1 for1 ǫ t. • − ≤ 2 Basic Constructions of Homotopy Theory in FRL In this section, we define the fundamental notions of homotopy theory in the category FRL, such as the homotopy relation and the mapping cylinder. We begin with an overview of our approach to homotopy in FRL, and then discuss alternate Fr¨olicher structures on the unit interval which are used in this and subsequent sections. 2.1 Our Approach to Homotopy Theory in FRL One might begin investigating homotopy theory in FRL by simply following the homotopy theory of topological spaces, replacing continuous functions with smooth ones. One can certainly define the notion of a homotopy H : I X Y between smooth maps H(0, ) and H(1, ) in this way (which we do).× One→ can even get as far as the left Puppe− sequence− (see [4]), but eventually difficulties begin to arise. Extending functions defined on a subspace of a Fr¨olicher space tends to be a little tricky, and so the definition of a cofibration in FRL is one that needs careful consideration. We envisage to construct the right Puppe sequence in a future paper. To do this we define a slightly weaker notion of cofibration than the notion obtained from topological spaces. In addition, we define the mapping cylinder of a smooth map f : X Y using not the unit interval, but a modified version called the weakly flattened→ unit interval, denoted I, which, as one can show, is topologically homeomorphic to the unit interval. This modified structure on the unit interval allows us to show that the inclusion of a space X into the mapping cylinder of f : X Y is a cofibration (in our weaker sense ). → 4 The weakly flattened unit interval is useful, but it also has its drawbacks. It would be ideal to have a single structure on the unit interval that can be used throughout out homotopy theory, but the weakly flattened unit interval is not suitable, because it has the rather restrictive property that a smooth map f : I I on the usual unit interval often does not define a smooth map f : I I unless→ the endpoints of the interval are mapped to the endpoints. This restrictive→ property means that we only use the flattened unit intervals where they are absolutely necessary. In our future work, we will investigate whether with our modified notions of cofibration and mapping cylinder, Baues’ cofibration axioms are satisfied. 2.2 Flattened Structures on the Unit Interval We define two main Fr¨olicher structures which we call the flattened unit in- terval and the weakly flattened unit interval . Let ( I , I ) be the subspace structure induced on I by the inclusion I ֒ R. C F → Definition 2.1 The Fr¨olicher space (I, I, I), where the structure ( I, I) is the structure generated by the set C F C F 1 F = f I there exists 0 <ǫ< with f(t)= f(0) for t [0,ǫ) and { ∈F | 4 ∈ f(t)= f(1) for t (1 ǫ, 1] , ∈ − } is called the flattened unit interval. It is easy to see that any continuous map c : R [0, 1] defines a structure curve on I if and only if it is smooth at every point →t R, where c(t) (0, 1), . We define the left (resp. right) flattened unit∈ interval, denoted∈ by I− (resp. I+), to be the Fr¨olicher space whose underlying set is the unit interval [0, 1], and structure is the structure generated by the structure functions in I that are constant near 0 (resp. 1). F Definition 2.2 The Fr¨olicher space (I, I, I), with the structure defined below is called the weakly flattened unit intervalC F . The underlying set is the unit interval; the structure ( I, I) is generated by the family C F dn dn F = f I lim f(t)=0, lim f(t)=0, n 1 . { ∈F | t→0+ dtn t→1− dtn ≥ } We call the property, for all f F , ∈ dn dn lim f(t)=0, lim f(t)=0, n 1, t→0+ dtn t→1− dtn ≥ the zero derivative property of f. We shall prove that all structure functions on I have the zero derivative property, in other words, I = F . To that effect, we need the following lemma. F 5 Lemma 2.1 Let c : R R be a smooth real-valued function at t = t0, and let f : R R be a smooth→ real-valued function at t = c(t ). Then, → 0 dn (f c)(t )= f (n)(c(t ))(c′(t ))n + terms of the form dtn ◦ 0 0 0 (k) ′ m1 ′′ m2 (n−1) mn−1 af (c(t0))(c (t0)) (c (t0)) ... (c (t0)) , where k Proof. The proof is done by induction. For the sake of brevity, we call the (n) ′ n term f (c(t0))(c (t0)) the primary term for n, and the terms of the form (k) ′ m1 ′′ m2 (n−1) mn−1 af (c(t0))(c (t0)) (c (t0)) ... (c (t0)) the lower order terms for n. The statement is true for n = 1 and for n = 2. Suppose the result is true for n = k. To show that the result holds for n = k + 1, since dk+1 d (f c)(t )= (f (k)(c(t ))(c′(t ))k) dtk+1 ◦ 0 dt 0 0 d (j) ′ m1 ′′ m2 (k−1) mk−1 +terms of the form dt (af (c(t0))(c (t0)) (c (t0)) ... (c (t0)) ), where j < k + 1 and a R, we need only show that ∈ d (af (j)(c(t ))(c′(t ))m1 (c′′(t ))m2 ... (c(k−1)(t ))mk−1 ) dt 0 0 0 0 gives rise to lower terms for n = k + 1, which is by the way straightforward. dn dn Theorem 2.1 I = f I lim + n f(t) = 0 = lim − n f(t) =: F F { ∈F | t→0 dt t→1 dt } Proof. That F I is evident. We must show the reverse inequality. Let 1 ⊆ F 0 <ǫ< , and 0 cM (t)=(1 αǫ( t ))βM (t)+ αǫ( t ), − | | | | where αǫ : R R is a smooth braking function as defined in the Preliminaries, → and βM : R R is given by → Mt if t 0 βM (t)= − ≤ t if t> 0 It is easily seen that cM is continuous over all R, and smooth over all R except at t = 0. Also note that 0 6 For n> 1, we have dn dn cM (t)= βM (t)=0, for t ( ǫ, 0) (0,ǫ). dtn dtn ∈ − ∪ We now show that for cM ΓF . To this end, let f F . To show that ∈ ∈ f cM : R R is smooth, it is obvious that we need only concentrate on the point◦ t = 0,→ because f c is smooth at every t = 0. It follows for t = 0, and ◦ 6 + 6 n N that Lemma 2.1 applies. But as t 0, cM (t) 0 , and so, letting ∈ → → s = cM (t), we have (j) (j) lim f (cM (t)) = lim f (s)=0, t→0 s→0+ for all j N, by the zero derivative property of f. Thus, as t approaches ∈ dn the value 0, the primary term and all the lower order terms of n (f cM )(t) dt ◦ vanish, and we have shown that f cM is smooth at t = 0. This implies that ∞ ◦ f cM C (R, R) for all f F . It follows that cM ΓF . ◦ ∈ ∈ ∈ We are now ready to show that I F . To this end, suppose that we F ⊆ are given a structure function f I. We shall show that this f has the zero derivative property, and is thus an∈ elementF of F . Since f I, we know that f c is a smooth real-valued function for every ∈F ◦ c ΓF . In particular, f cM is smooth for all 0 − + As t 0 , cM (t) 0 ; let us consider the lower order terms for n. Each term of the→ form → (k) ′ m1 ′′ m2 (n−1) mn−1 af (cM (t))(cM (t)) (cM (t)) ... (cM (t)) (i) mi (i) has some term (cM (t)) , for some i> 1, with mi = 0. But limt→0− cM (t) = 0, if i> 1, and so 6 (k) ′ m1 ′′ m2 (n−1) mn−1 lim af (cM (t))(cM (t)) (cM (t)) ... (cM (t)) =0. t→0− So all the lower order terms fall away, therefore n d (n) ′ n limt→0− dtn (f cM )(t) = limt→0− f (cM (t))(cM (t)) ◦ (n) n = limt→0− f (cM (t))( M) (n) −n = lim + f (s)( M) , s→0 − where s = cM (t). In a similar way one shows that n d (n) lim (f cM )(t) = lim f (s). t→0+ dtn ◦ s→0+ (n) n (n) But f cM is smooth, therefore lims→0+ f (s)( M) = lims→0+ f (s), which ◦ (n) − implies that lims→0+ f (s)=0. 7 We have shown that the zero derivative property of f holds for the left endpoint of the unit interval. To show that the zero derivative property of f holds for the right endpoint of f, note that dM : R R, dM (t)=1 cM (t), is → − a smooth real-valued function with d(0) = 1, and 0 dM (t) 1 for all t R. ≤ ≤ ∈ One can follow a similar procedure to the above, using dM instead of cM to (n) show that lims→1− f =0. 2.3 Some Properties of Smooth Functions between the Flattened Unit Intervals One has to be careful when dealing with the various flattened unit intervals. A smooth function f : I I from the R- Fr¨olicher subspace unit interval I to itself need not define a→ smooth function f : I I, for example. Conversely, not every smooth function f : I I defines a smooth→ function f : I I. In particular, we need to be aware of→ the fact that addition and multiplic→ation of functions when defined between the various flattened unit intervals does not preserve smoothness, as is the case with the usual unit interval. Example 2.1 1 The function f : I I, f(t) = 2 t is clearly smooth, but the corresponding function f : I I, given→ by the same formula, is not smooth. To see this, let α : R R be a→ smooth braking function with the properties that → α(t)= 1, for t< 3 , • − − 4 α(t)= t, for 1 Example 2.2 8 The function f : I I, f(t)= √t, is smooth, but the corresponding f : I I, given by the same→ formula, is not smooth. This follows from the fact that→f is smooth on the open interval (0, 1), and a generating function g on I is constant near 0 and 1. On the side, f : I I is not smooth, because if c : R I is a structure curve with c(t)= t2 near→t = 0, then (f c)(t)= t near t =→ 0, which is not smooth on I at t = 0. ◦ | | Example 2.3 The functions f,g : I− I−, given by f(t) = 1 √t and g(t) = 1 are both → 2 4 1 √ 1 smooth, but the sum f(t)+ g(t)= 2 t + 4 is not smooth. The following lemma follows from the definition of the Fr¨olicher structures on the various flattened unit intervals. Lemma 2.2 Let f : I I be a smooth function with the properties that f(0) = 0 and f(1) = 1. Then the→ following maps are smooth: f : I I±, • → f : I I, • → f : I± I, • → f : I I, • → f : I I. • → The function defined in the following example is for later reference. Example 2.4 Let H : I I− I− be given by H(t,s) = (1 α(t))s, where α : R R is a smooth braking× → function with the properties that− → α(t)=0 for t< 1 , • 4 0 α(t) 1 for all t R, • ≤ ≤ ∈ α(t)=1 for t> 3 . • 4 We show that H is smooth. To see this, let f : I− R be a generating function on I−. So f is constant near 0. Now, let c : R →I I− be a structure curve, given by c(v) = (t(v),s(v)). The curve t is a structure→ × curve on I, and so is a smooth real-valued function for all v R, except possibly when t(v)=0or t(v) = 1. Similarly, the curve s is a structure∈ curve on I−, and so is smooth for all v R except possibly when s(v) = 0. Now consider the composite H c : R∈ I−. Clearly, α(t(v)) is smooth for all v, since the only possible points◦ for→ non-smoothness occur when t(v) = 0or t(v) = 1, and α(t(v)) is locally constant near these points. Consequently, H c is smooth everywhere except possibly when s(v) = 0. Now, let’s consider f ◦ H c : R R; the only possible points for non-smoothness are those in which ◦s is 0,◦ i.e. H→ = 0. But f is a structure generating function on I−, and so is locally constant◦ near 0. This shows that f H c is smooth for all v R, and thus H is smooth. ◦ ◦ ∈ 9 2.4 Homotopy in FRL and Related Objects Definition 2.3 (1) Let X be a Fr¨olicher space, and x0, x1 X. We say that x is smoothly path-connected to x if there is a smooth path∈ c : I X such 0 1 → that c(0) = x0 and c(1) = x1. We write x0 x1. The relation is called smooth homotopy when it is applied to hom-sets.≃ ≃ (2) Let f : X Y be a map of Fr¨olicher spaces. f is called a smooth homotopy equivalence→ provided there exists a smooth map g : Y X such that → f g 1Y and g f 1X . ◦ ≃ ◦ ≃ One can show that smooth homotopy is a congruence in RFL. In practice, we say that smooth maps f,g : X Y are smoothly homotopic if there exists a smooth map H : I X Y with→ H(0, )= f and H(1, ) = g. If A X is subspace of X, then× we→ say that H is a− smooth homotopy− (rel A) if the⊆ map H has the additional property that H(t,a) = a for each t I and a A. See Cherenack [5] and Dugmore [6] for more detail regarding smooth∈ homotopy.∈ The notion of deformation retract is fundamental to topological homotopy theory. The following definitions are adapted for smooth homotopy, and will be needed at a later stage. Definition 2.4 Let A X be a subspace of a Fr¨olicher space X, and let i : A ֒ X denote the inclusion⊆ map. Then → We say that A is a retract of X if there exists a smooth map r : X A • → such that ri =1A. We call r a retraction. We call A a weak deformation retract of X if the inclusion i is a smooth • homotopy equivalence. The subspace A is called a deformation retract of X if there exists a re- • traction r : X A such that ir 1X . → ≃ The subspace A is called a strong deformation retract of X if there exists • a retraction r : X A such that ir 1X (relA). → ≃ Definition 2.5 The mapping cylinder If of f : X Y is defined by the fol- lowing pushout → f X / Y i1 I X / If × where i : X I X is given by i (x)=(1, x), for any x X. We denote the 1 → × 1 ∈ elements of If by [t, x] or [y], where (t, x) I X and y Y . Replacing I X in the above pushout diagram∈ × by I X∈ or I X, we obtain × × × the flattened mapping cylinder If and weakly flattened mapping cylinder If of f respectively. We use the same notation for elements of these flattened mapping cylinders as described above for the mapping cylinder. 10 There is also a map i0 : X I X, defined by i0(x) =(0, x) for x X. This →′ × ∈ induces an inclusion map i0 : X If , which identifies X with the Fr¨olicher ′ → subspace i0(X) of If . An inclusion is induced in a similar way for the flattened mapping cylinders. If one identifies 0 X to a point in the mapping cylinder { }× If of a map f : X Y , then one obtains the mapping cone Tf of the → map f. In a similar fashion, we define the flattened mapping cone Tf and weakly flattened mapping cone Tf of a smooth map f : X Y . → 2.5 Cofibrations in FRL A cofibration is a map i : A X for which the problem of extending functions from i(A) to X is a homotopy→ problem. In other words, if a map f : i(A) Z can be extended to a map f ∗ : X Z, then so can any map homotopic to f.→ For topological spaces, the usual definition→ is phrased in a slightly more restrictive way. The extension of a map g H f, for some homotopy H : I i(A) Z, is required to exist at every level of≃ the homotopy simultaneously. In× other→ words, one requires each H(t, ) to be extendable in such a way that the resulting homotopy H∗ : I X −Z is continuous. We weaken this× definition→ somewhat, to enable smooth homotopy extensions to be more easily constructed using a flattening at the endpoints of the homo- topy. This enables us to characterize smooth cofibrations in terms of a flattened unit interval, and then later to relate smooth cofibrations to smooth neigh- borhood deformation retracts. Our definition of smooth cofibration, though different from from Cap’s definition, see [1], leads to several classical results as does Cap’s. As pointed out by Cap, the analogue of the classical definition of cofibration would not allow even 0 ֒ I to be a smooth cofibration. So, we have the following { } → Definition 2.6 A smooth map i : A X is called a smooth cofibration if, corresponding to to every commutative→ diagram of the form i f A / X / Z , mmm6 mmm (0,1A) mmm mmm G mm I A × there exists a commutative diagram in FRL of the form f X / Z , tt: O F tt (0,1X ) tt G′ tt tt I X o I A × 1I ×i × ′ ′ 1 where G : I A Z is given by G (t,a) = G(αǫ(t),a) for some 0 <ǫ< 2 , and each t ×I, a →A. ∈ ∈ 11 The problem of extending a map smoothly from a subspace of a Fr¨olicher space to the whole space is a more difficult problem than simply extending con- tinuously. It is mainly for this reason that the definition of smooth cofibration differs somewhat from the corresponding definition of a topological cofibration. Lemma 2.3 Let i : A X be a smooth cofibration, then i is an initial mor- phism in FRL. In addition,→ if A is Hausdorff, then i is injective. So in this case A can be regarded as a subspace of X. Proof. Let us show that every smooth map f : A R factors through i, that → is for every f A, there exists f˜ X such that f = f˜ i. To this end, consider the smooth∈ F map G : I A ∈ FR, given by H(t,a) =◦ tf(a). Clearly, × → 0 A = G(0, ), where 0 : X R is the constant map 0. It follows that there is map| F : I −X R such that→ F (1 i)= G′. Then, clearly f˜ := F (1, ) has the desired× property.→ ◦ × − The remaining part of the proof of Proposition 3.3, in [1], holds verbatim here as well. In this paper, we are interested only in cofibrations that are injective. Hence- forth, all cofibrations are assumed to be injective. All topological cofibrations are inclusions, and this result is true for smooth cofibrations too. The proof of the following lemma is essentially the same as the proof given by James [9] for the topological result, although James’s proof is in some sense dual to ours, using path-spaces in place of cartesian products and the adjoint versions of our homotopies. Lemma 2.4 A cofibration i A / / X is a smooth inclusion. Proof. Let Ii be a mapping cylinder of i, and let j : X Ii be the standard inclusion map. Consider the smooth map γ : I I, γ(t)=1→ t, for all t I, → − ∈ and the quotient map q : (I A) X Ii; we have the following commutative diagram × ⊔ → i j A / / X / Ii , mmm{=6 mmm {{ (0,1A) mm { mmq { mmm {{ m {{ I A {{ × {{ G {{ γ×1A {{ {{ I A × where G(t,a) = [(1 t,a)]. Notice that the map G is smooth. Since i is a − 12 cofibration, we have the commutative diagram j X / Ii , uu: O F uu (0,1X ) uu G′ uu uu I X o I A × 1I ×i × ′ 1 I where G (t,a) = G(αǫ(t),a) for some 0 <ǫ< 2 . Define U : X i by U(x) = F (1, x). We have U i = G′(1, ), where G′(1,a) = [(0,a)], for→ every a A. Thus the assignment ◦a G′(1,a−) defines the usual inclusion of A into the∈ mapping cylinder. From this7→ we deduce that U i is an inclusion, and hence i is an inclusion. ◦ There is an equivalent formulation of definition 2.6, given in the following lemma. Lemma 2.5 A smooth map i A / / X is a cofibration if and only if, for every smooth map h : (0 X) (I− i(A)) Z, the following diagram × ∪ × → h (0 X) (I− i(A)) / Z , × ∪ × ooo7 ooo j ooo ooo G oo I− X × where j is the evident inclusion, exists in FRL. i Proof. Suppose that the inclusion A / / X is a smooth cofibration, and suppose that h : (0 X) (I− i(A)) Z is a smooth map. We have the diagram × ∪ × → h (0 B) (I− i(A)) / Z . × ∪ × j I− X × We need to fill in a smooth map G : I− X Z which makes the resulting × → diagram commute. To do this, notice that h I−×i(A) is smooth, and thus the corresponding map h I i(A), using the usual| unit interval, is also smooth. We have the following diagram| × i h|0×X A / / X m/6 Z , mmm mmm (0,1A) mmm mm hI−×A mm I A × 13 where h 0×X (0, ): X Z is denoted as h 0×X . The fact that i is a smooth cofibration| yields− the following→ FRL-commutative| diagram: h0×X X / Z , tt: O F tt ′ (0,1A) tt (h|I−×A) tt tt I X o I A × 1I ×i × 1 − ′ − where (h I ×A) (t,a) = h I ×A(αǫ(t),a), for some 0 <ǫ< 2 . Now, chose a smooth braking| function β| : R R with the following properties. → α(t)=0 for t< ǫ , • 2 α(t)= t for ǫ i f A / X / Z . mmm6 mmm (0,1A) mmm mmm G mm I A × There exists the diagram i f A / X / Z , mmm6 mmm (0,1A) mmm mmm G′ mm I A × ′ where G (t,a)= G(αǫ(t),a). Our hypothesis allows us to construct the diagram f∪G′ (0 X) (I− i(A)) / Z . × ∪ × ooo7 ooo j ooo ooo H oo I− X × 14 ′ Note that f G is smooth since αǫ(t) is constant near 0. Since H is smooth on I− X it∪ defines a smooth map on I X. One can verify that the diagram × × f X / Z tt: O H tt (0,1X ) tt G′ tt tt I X o I A × 1I ×i × commutes as required. 3 Smooth Neighborhood Deformation Retracts This section is concerned with the formulation of a suitable notion of smooth neighborhood deformation retract. For topological spaces, the statement that a closed subspace A of X is a neighborhood deformation retract of X is equivalent to the statement that the inclusion i : A ֒ X is a closed cofibration. We show that in the category of Fr¨olicher spaces there→ is a notion of smooth neighborhood deformation retract that gives rise to an analogous result that a closed Fr¨olicher subspace A of the Fr¨olicher space X is a smooth neighborhood deformation retract of X if and only if the inclusion i : A ֒ X comes from a certain subclass of cofibrations. As an application, we construct→ the right Puppe sequence. 3.1 SNDR pairs and SDR pairs The definition of ‘smooth neighborhood deformation retract’ that we adopt in this paper is similar to the definition of ‘R-SNDR pair’suggested in [6], but we have modified the definition in order to retain only the essential aspects of ‘first coordinate independence’ defined in [6]. We begin by defining the ‘first coordinate independence property’ of a func- tion on a product of a Fr¨olicher space with I (or I−, I+). Definition 3.1 Let i : A X be a smooth map, and c : R X a structure curve on X. Define → → −1 Λ(c,i) = t∗ c (i(A)) there exists a sequence tn of real numbers { ∈ | −1 { } with limn tn = t and each tn c (X i(A)) . →∞ ∗ ∈ − } The points in Λ(c,i) are those values in R where the curve ‘enters’ i(A) from X i(A), or ‘touches’ a point in i(A) whilst remaining in X i(A) nearby. Now,− we are ready to define the ‘first coordinate independence− property’ for a structure function on a product. Definition 3.2 Let i : A X be a smooth map and suppose f : I X R is a structure function on I →X. Let c : R I X, given by c(s) =× (t(s→), x(s)) have the following properties× → × 15 The map x(s) is a structure curve on X. • For all ǫ> 0, t(s) is a smooth real-valued function on R [s • − ∪s∗∈Λ(x,i) ∗ − ǫ,s∗ + ǫ]. If, for every such map c, the composite f c is a smooth real-valued function, then we say that f : I X R has the first◦ independence property (FCIP) with respect to i. × → Extending the definition, we say that a map g : I X Y has the FCIP with respect to i if the composite h g : I X R has× the→ FCIP with respect ◦ × → to i for every h Y . ∈F Notice that we can formulate a similar definition of the FCIP if we replace I throughout by I− or I+, leaving the rest of the definition unchanged. We will have occasion to use this type of first coordinate independence property in the later part of this work. Note. Let i : A X, and suppose that we are given a map g : I X Y . Let f : Y R be a→ structure function on Y , and suppose that f g×: I →X R has the→ FCIP with respect to i for any such f. Then, given◦ a smooth× → map h : Y Z, the composite f ′ h g : I X R has the FCIP with respect to i for any→ structure function f◦′ on◦Z. × → The above note applies equally well if g : I− X Y or g : I+ X Y has the FCIP with respect to i when composed with× a→ smooth function× h on→Y . Example 3.1 1. For any i : A X, the projection onto the second coordinate πX : I X X has the FCIP. → × → 2. Let α : R R be a smooth braking function with the properties that → α(t) = 0 if t< 1 , • 4 0 < α(t) < 1 if 1 t 3 , • 4 ≤ ≤ 4 α(t) = 1 if 3 16 H(1, x) i(A) for all x X with u(x) < 1, • ∈ ∈ then the pair (X, A) is called a smooth neighborhood deformation retract pair, or SNDR pair for short. If, in addition, H is such that H(1 X) i(A), then the pair (X, A) is called a smooth deformation retract pair,× or an⊂ SDR pair for short. The subspace A is called a smooth neighborhood deformation retract or smooth deformation retract of X if (X, A) is an SNDR pair or SDR pair, respectively. The pair (u,H) is called a representation for the SNDR (or SDR) pair. Example 3.2 1. The pair (X, ) is an SNDR pair. A representation is u(x) = 1, H(t, x)= x, for each t I and∅ x X. 2. The pair∈ (X,X) is∈ an SNDR pair. A representation is u(X) = 0, H(t, x)= x, for each t I and x X. ∈ ∈ Lemma 3.1 The pair (I−, 0) is an SDR pair. Proof. Let α : R R be the smooth braking function of Examples 3.1. A representation for (→I−, 0) as an SDR pair is (u,H), where u : I− I and H : I I− I− are given by u(s)= s, and H(t,s)=(1 α(t))s. Clearly,→ the identity× u :→I− I is smooth. And the map H, as shown− in Example 2.4, is smooth and clearly→ has the FCIP with respect to the inclusion, since whenever v approaches a value for which s(v) = 0, one has g((1 α(t(v)))s(v)) = g(0) − for v in a neighborhood of this value and g I− . ∈F Lemma 3.2 The pair (I, 0, 1 ) is an SNDR pair. { } Proof. A representation (u,H) for the SNDR pair can be given as follows. Define u : I I to be a bump function such that → u(t)=0 for t =0 or t = 1, • u(t)=1 for t [ 1 , 3 ], • ∈ 4 4 0 17 where t = 0, 1 and s = 0, 1. The braking function αǫ ensures that H is locally constant in the tb variable whenever t is near 0 or 1, so no problem arises from the t component. When s is near s = 0, we have H(t,s) near 0, and so the generating function f is locally constant. Similarly, when s is near s = 1, we have H(t,s) near 1, and the generating function f is again locally constant. We now show that the product of SNDR pairs is again an SNDR pair. (Theorem 3.1 Let i : A ֒ X and j : B ֒ Y be inclusion mappings. If (X, A and (Y,B) are SNDR pairs,→ then so is → (X Y, (X B) (A Y )). × × ∪ × If one of (X, A) or (Y,B) is an SDR pair, then so is the pair (X Y, (X B) (A Y )). × × ∪ × Proof. Let α : R I be a smooth braking function with the properties that 1→ 3 R R α(t) = 0 for t 4 , and α(t) = 1 for t 4 , and let β : be a smooth ≤ ≥ → 1 increasing braking function with the properties that β(t) = t for t 4 , and 3 ≤ β(t) = 1 for t 4 . Suppose that (u,H) and (v,J) are representations for the SNDR pairs (X,≥ A) and (Y,B), respectively. Let u : X I, and v : Y I be given by u(x)= β(u(x)) and v(y)= β(v(y)) respectively.→ Define w : X →Y I by w(x, y) = u(x)v(y). The braking function β ensures smoothness of× u →and v, and consequently of w. We have w−1(0) = (X B) (A Y ), as required. Define Q : I X Y X Y as follows . × ∪ × × × → × (H(α(t), x),J(α(t),y)) if u(x)= v(y)=0 u(x) Q(t,x,y)= (H(α(t), x),J(α( )α(t),y)) if v(y) u(x), v(y) > 0, v(y) ≥ (H(α( v(y) )α(t), x),J(α(t),y)) if u(x) v(y), u(x) > 0. u(x) ≥ We must show that Q is a smooth map, with the first coordinate independence property with respect to the inclusion (X B) (A Y ) ֒ X Y . We first consider each part of the definition of Q ×separately.∪ × The→ first part× is clearly smooth. Let us verify that Q is smooth on the second part of its definition; the third part is similar. u(x) We need only focus on the component J(α( v(y) )α(t),y). Each function u(x) making up J(α( v(y) )α(t),y) is smooth individually, so we need only pay extra attention to those parts that involve flattened unit intervals, remembering that addition and multiplication on the flattened unit interval need not preserve smoothness, as is the case for the usual unit interval. u(x) u(x) So let us consider α( v(y) ); it is smooth except possibly when v(y) approaches 0 or 1, since it is here that structure curves on the flattened unit interval need not be smooth in the usual sense. Clearly, if u(x) approaches 0 and v(y) does u(x) not approach 0, then the braking function α ensures that v(y) = 0 near such points. If v(y) approaches 0, then u(x) must approach 0 too. This situation is dealt with later. 18 Thus, Q, in part two of the definition, is smooth, and one can show similarly that Q in the third part of the definition is smooth as well. Let us now consider the overlaps of the three parts of the definition of Q. Observe that if u(x) is in a sufficiently small neighborhood of v(y), with u(x) =0 6 and v(y) = 0, then we have α( u(x) = α( v(y) ) = 1, and so the second and third 6 v(y) u(x) parts of the definition of Q coincide here. Thus, it remains only to show that Q is smooth as u(x) and v(y) both approach 0. If Q is smooth in each of its coordinates then it is smooth, so consider the coordinate involving the map J. Let c : R I X Y be a structure that is given by c(s) = (t(s), x(s),y(s)). Then, the→ map×c :×R I Y , given by 1 → × (α(t(s)),y(s)) if u(x(s)) = v(y(s))=0 u x s c (s)= (α( ( ( )) )α(t(s)),y(s)) if v(y(s)) u(x(s)), v(y(s)) > 0 1 v(y(s)) ≥ (α(t(s)),y(s)) if u(x(s)) v(y(s)), u(x(s)) > 0 ≥ is a map satisfying the conditions of Definition 3.2, since its second coordinate is smooth, but its first coordinate may be singular as v(y(s)) ( and hence u(x(s))) approaches 0. Since J has the first coordinate independence property, the map J(α(t(s)),y(s)) if u(x(s)) = v(y(s))=0 (Joc )(s)= J(α( u(x(s)) )α(t(s)),y(s)) if v(y(s)) u(x(s)), v(y(s)) > 0 1 v(y(s)) ≥ J(α(t(s)),y(s)) if u(x(s)) v(y(s)), u(x(s)) > 0 ≥ is smooth. Thus, Q c is smooth, and since c is arbitrary, Q is smooth. In a similar way, the coordinate◦ of Q involving H can be shown to be smooth. We now verify that Q satisfies the required boundary conditions. When t = 0, all three lines defining Q reduce to (H(0, x),J(0,y)) = (x, y). Let x A and y B; then u(x)= v(y) = 0. Therefore, Q reduces to (H(α(t), x),J(α(∈t),y)) = (x,∈ y). If x A and y / B, then Q is given by the second part of its definition, which reduces∈ to (H(∈α(t), x),J(0,y)). The case when x / A and y B is similar. If t =1 and 0 < w(x, y) < 1 then either 0 < u(x) <∈1 or 0 < v(∈y) < 1. Suppose that 0 < u(x) < 1. Then either u(x) v(y) or v(y) < u(x). If u(x) v(y), then Q is given by the second part of≤ its definition, which reduces ≤ to (H(1, x),J(α( u(x) ,y)) i(A) Y . If v(y) < u(x), then the third part of the v(y) ∈ × definition of Q applies and Q reduces to (H(α( v(y) ), x),J(1,y)) X j(B). u(x) ∈ × Finally, we must show that for any f X×Y , f Q has the first coordinate . independence property with respect to the∈F inclusion (◦X B) (A Y ) ֒ X Y To this end, consider a map c : R I X Y , given by×c(s)∪ = (t×(s), x→(s),y(×s)). → × × Let sn be a sequence of real numbers converging to s with c(sn) (X Y ) { } ∗ ∈ × − ((A Y ) (X B)), and c(s∗) (A Y ) (X B). There are three cases to consider.× ∪ × ∈ × ∪ × Suppose that c(s∗) A B. Then x(s∗) A and y(s∗) B. The fact that • H and J have the first∈ × coordinate independence∈ property∈ with respect to i and j respectively means that each coordinate of Q is smooth, and so Q is smooth. 19 Suppose that c(s ) A Y , and that y(s ) / B. Then at each of the • ∗ ∈ × ∗ ∈ points c(sn), (Q c)(sn) is given by the second part of the definition of ◦ Q, for n large enough. Since x(s∗) A, the component of Q involving H is smooth, since H has the first coordinate∈ independence property. For u(x(s)) any s in a neighborhood of s∗, α( v(y(s)) ) = 0. Thus, the component of Q involving J is constant for s in a neighborhood of s∗, and so is smooth there. The case with c(s∗) X B, and x(s∗) / A is similar to the second case • above. ∈ × ∈ For the last part of the theorem, suppose that (u,H) represent (X, A) as an ′ 1 ′ SDR pair. If we replace u by u = 2 u, then (u ,H) also represent (X, A) as an SDR pair. Making the above constructions now with u′ in place of u, it follows that w(x, y) < 1 for all (x, y) and so Q(1,x,y) (X B) (A Y ). This completes the proof. ∈ × ∪ × 4 Cofibrations In this section, we show that for a subspace A X that is closed in the under- lying topology, the inclusion i : A X is a cofibration⊆ if and only if (X, A) is an SNDR pair. → Definition 4.1 Let i : A X be a cofibration. We call i a cofibration with FCIP if any homotopy extension→ can be chosen to have the FCIP with respect to i. Using the equivalent formulation of the notion of cofibration, given by Lemma 2.5, we may restate Definition 4.1 as follows: A cofibration i : A X is a cofibration with the FCIP if and only if the map G that we may→ fill in to complete the commutative diagram h (0 X) (I− A) / Y × ∪ × p7 p p j p p p G p I− X × may be chosen to have the FCIP with respect to the inclusion i. We have the following result, which corresponds to a similar topological result. Lemma 4.1 A smooth map i : A X is a cofibration (with the FCIP) if and only if (0 X) (I− A) is a→ retract of I− X, (where the retraction r : I− X (0× X∪) (I−× A) has the FCIP ). × × → × ∪ × 20 Proof. In the one direction, suppose that (0 X) (I− A) is a retract of I− X. We wish to complete the following diagram:× ∪ × × h (0 X) (I− A) / Y . × ∪ × p7 p p j p p p G p I− X × By hypothesis, there exists r : I− X (0 X) (I− A) such that r j = 1. Define G = h r. If r has the FCIP,× then→ so× does∪h r×. ◦ Conversely,◦ suppose that i : A X is a cofibration◦ (with the FCIP). We may find a map r such that the diagram→ 1 (0 X) (I− A) / (0 X) (I− A) × ∪ × j ×j5 ∪ × j j j j j r j j I− X × commutes. Thus, r j =1. If i is cofibration with the FCIP with respect to i, then r can be chosen◦ to have the FCIP. The next theorem shows the relationship between cofibrations, retracts and SNDR pairs. Theorem 4.1 Let i : A X be an inclusion, with A closed in the underlying topology of X. Then the→ following are equivalent. (1) The pair (X, A) is an SNDR pair. (2) There is a smooth retraction r : I− X (0 X) (I− A) with the FCIP. × → × ∪ × (3) The map i : A X is a cofibration with the FCIP. → Proof. To show that (1) and (2) are equivalent, note that the pair (I− X, (0 X) (I− A)) is an SDR pair, as a consequence of Lemma 3.1 and× Theorem× 3.1.∪ Let (w,Q× ) be a representation for the pair (I− X, (0 X) (I− A)) as an SDR pair, and let Q be constructed as in Theorem× 3.1.× Define∪ × r : I− X (0 X) (I− A) × → × ∪ × by r(t, x)= Q(1,t,x), where (t, x) I− X. We observe that r has the FCIP, since Q has this property, and Q has∈ this× property since each of its components has this property. The equivalence of (2) and (3) is Lemma 4.1. We need only show that (2) implies (1). Let r : I− X (0 X) (I− A) be a retraction with the FCIP with respect to i. Define× →H ×: I ∪X × X × → by H(t, x) = (πX r)(α(t), x), where πX is the projection onto the second ◦ 21 coordinate, and α : R R is a braking function with the following properties: → 3 3 α(t) = 0 for t 0, α(t) = 1 for t 4 , and 0 < α(t) < 1 for 0 Now, define u : X I by → 1 β(α(t) (πI r)(1, x)(πI r)(α(t), x))dt u(x)= 0 − ◦ ◦ . R 1 0 β(α(t))dt R It is clear that u is a smooth mapping. We now verify that (u,H) represents (X, A) as an SNDR pair. (1) Let x A. Clearly, (πI r)(1, x) = 1 and πI r)(α(t), x) = α(t), and so 1 ∈ ◦ ◦ β(α(t) (πI r)(1, x)(πI r)(α(t), x))dt = 0. Thus, u(x) = 0, for all x A. 0 − ◦ ◦ ∈ R (2) Suppose that x X A. Since 0 (X A) is open in the underlying topology on (0 X) (I− ∈A), we− may choose× an− open neighborhood W 0 (X A) of (0,× x). Since∪ r ×is continuous, there is a neighborhood V I− ⊆ X×such− that ⊆ × r(V ) W 0 (X A). Now, consider the mapping qx : I I X, given by ⊆ ⊆ × − → × qx(t) = (α(t), x), for each x X. This is clearly smooth. Thus, there exists a − ∈ neighborhood U I such that qx(U) V . In other words, U x V . So, ⊆ ⊆ ×{ }⊆ we have (πI r)(α(t), x) = 0, for all t U. Thus, we have ◦ ∈ β(α(t) (πI r)(1, x)(πI r)(α(t), x))dt + β(α(t))dt u(x)= I−U − ◦ ◦ U > 0. R 1 R 0 β(α(t))dt R Combining this with part (1), we deduce that u−1(0) = A. (3) Suppose that x is such that u(x) < 1. There must be a neighborhood U of I such that (πI r)(1, x)(πI r)(α(t), x) > 0, for t U. Thus (πI r)(1, x) > 0, but this implies that◦ r(1, x) ◦I A, and hence H(1∈, x) A. The proof◦ is complete. ∈ × ∈ 22 5 The Mapping Cylinder In this section we show that the inclusion of X into the flattened mapping cylinder If of a map f : X Y is a cofibration with the FCIP. → Theorem 5.1 Let f : X Y be a smooth map. Then, the pair (If ,X) is an SNDR pair. → Proof. Let α : I R be a smooth braking function with the following proper- ties: α(t) = 0 if 0 →t 1 , α(t) = 1 if 3 t 1, 0 < α(t) < 1, otherwise. Define ≤ ≤ 4 4 ≤ ≤ two more braking functions α1, α2 : I R as follows: α1(0)= 0, 0 < α1(t) < 1 3 3 → 3 if 0 (I I X) (I Y ) × × ⊔ × ∼ where is the identification (t, 1, x) = (t,f(x)) for t I, and x X. Since H is smooth∼ when restricted to each component of the coproduct∈ (I ∈I X) (I Y ), × × ⊔ × H is smooth on the quotient I If . × We now verify that (u,H) is a representation for (If ,X) as an SNDR pair. u−1(0) = [0, x]= i (X). • 0 H(0, [t, x]) = [t, x] and H(0, [y]) = [y]. • H(s, [0, x]) = [0, x]. • If u[t, x] < 1, then t< 3 and so α (t) = 0. Thus, H(1, [t, x]) = [0, x]. • 4 2 This completes the proof. Finally, we have the following important corollary. Corollary 5.1 Given any smooth map f : X Y , the inclusion X ֒ If is a cofibration with the FCIP. → → 23 6 The Exact Sequence of a Cofibration Our aim in this section is to show how one can use SNDR pairs to prove the existence of the right exact Puppe sequence. We state the result in Theorem 6.1 and break the proof of the result up into a number of lemmas. We follow the method used by Whitehead [12] for the topological case. Throughout this section we work in the category FRL∗ of pointed Fr—’olicher spaces, and basepoint preserving smooth maps. Theorem 6.1 Let W be an object in FRL∗, and suppose that i : A ֒ X is a cofibration in FRL . For any basepoint x A X there is a sequence→ ∗ 0 ∈ ⊆ n ∗ n ∗ n ∗ n+1 (P k) n (P j) n (P i) n ... /[ A, W ] /[ Ti, W ] /[ X, W ] /[ A, W ] /... P P P P ∗ k∗ j i∗ ... /[ A, W ] /[Ti, W ] /[X, W ] /[A, W ] P which is an exact sequence in SETS , where j : X Ti is the inclusion dis- ∗ → cussed in Paragraphe 2.4 and k : Ti A is the quotient map defined below. → It is, in fact, possible to prove that theP sequence above is an exact sequence of groups as far as A, W ] and that the morphisms to this point are group homomorphisms, butP we shall not do so here. The reduced(flattened)suspension of a pointed Fr¨olicher space X is de- fined as X = (I/ 0, 1 ) X, X { } ∧ where the reduced join is defined as for topological spaces with the identified set taken as basepoint, and with 0 the basepoint of I. In this section, whenever we refer to the suspension of a space , we mean the reduced flattened suspension defined above. Lemma 6.1 If (x, A) is an SNDR pair and p : X X/A the quotient map, then the sequence → i p A / X / X/A is right exact. Proof. To show that the given sequence is right exact we must show that for any Fr¨olicher space W the following sequence is exact in SETS: ∗ p i∗ [X/A, W ] / [X, W ] / [A, W ] . It is easy to see that im p∗ ker i∗. To see the reverse inclusion, let g : X ⊆ → W be an element of [X, W ], with g A w0 (rel w0), where w0 W . Since i | ≃ ∈ A / X is an SNDR pair, the map i is a cofibration, and so we may extend ′ ′ ′ w0 to a smooth map g : X W such that g g. But g is constant on A, → ≃ ∗ ′ and so there exists a smooth map g1 : X/A W such that p (g1) = g . This shows that ker i∗ im p∗. → ⊂ 24 Lemma 6.2 For any smooth map f : X Y , the sequence → f l X / Y / Tf is right exact, where l is the usual inclusion of Y into the mapping cone; i.e. y [y] Tf . 7→ ∈ Proof. One can show that there is a homotopy commutative diagram f X / Y @@ AA @ AA l @@ j AA i @@ AA If p / Tf where i, j, and l are the usual inclusions, and p is the quotient map that collapses away 0 X to a point. Since, by Theorem 5.1, (If ,X) is an SNDR pair, it follows{ from}× Lemma 6.1 that the sequence i p X / If / Tf is right exact. It is fairly easy to show that j : Y If is a homotopy equivalence. Therefore, the sequence → f l X / Y / Tf is right exact. Lemma 6.3 For any smooth map i : A X, there is an infinite right exact sequence → n n n i i1 i2 i −1 i i +1 A / X / Ti / ... / Tin−2 / Tin−1 / ... where in, n 1, are inclusion maps. ≥ Proof. The pair (Ti,X) is an SNDR pair. The representation for the pair (If ,X) in Theorem 5.1 can be adapted to show this. One iterates the procedure of Lemmas 6.1 and 6.2. One can easily see that there is an isomorphism between Ti/X and A. Define q : Ti A to be the map which identifies X Ti to a point, followedP → ⊂ by the isomorphismP Ti/X A. → P Lemma 6.4 The sequence i1 q X X / Ti / A P is right exact. 25 Proof. As noted above the pair (Ti,X) is an SNDR pair. We have the com- mutative diagram i1 p X / Ti / Ti/X EE EE q E q0 EE E" A P where p : Ti Ti/X is the identification map, and q : Ti/X A is an → 0 → isomorphism. The top line of the diagram is right exact, by Lemma 6.1,P and so the sequence i1 q X / Ti / A P is right exact. There is a commutative diagram i1 i2 X / Ti / Ti1 CC CC q C q1 CC C! A P where q1 is a homotopy equivalence. ( See Whitehead [12] for more details of this map. ) Using commutative diagrams of this form, one can now proceed almost exactly as one does in the topological situation, as in Whitehead [12] for example, to get the following infinite right exact sequence: 1 i i1 q P i P i P q A / X / Ti / A / X / Ti / ... P P P n n P i P i1 ... / n A / n X / ... P P The definition of right exactness now gives us the exact sequence of Theorem 6.1. References [1] A. Cap, K-Theory for Convenient Algebra, Dissertationen, Faculty of Mathematics, University of Vienna, 1993. [2] Cherenack P., Applications of Fr¨olicher Spaces to Cosmology, Ann. Univ. Sci. Budapest 41(1998), 63-91. [3] P. Cherenack , Fr¨olicher versus Differential Spaces: A prelude to Cosmol- ogy, Kluwer Academic Publishers 2000, 391-413. 26 [4] P. Cherenack, The Left Exactness of the Smooth Left Puppe Sequence. In L. Tamassy and J. Szenthe, editors, New Developments in Differential Geometry, (Proceedings of the Colloquium on Differential Geometry, De- brecen, Hungary, July 26-30, 1994), Mathematics and Its Applications. Kluwer Academic Publishers, 1996. [5] P. Cherenack, Smooth Homotopy, Topology with Applications, (18):27-41, 1984. [6] B. Dugmore, The Right Exactness of the Smooth Right Puppe Sequence. Master’s Thesis, University of Cape Town, 1996. [7] A. Fr¨olicher, A. Kriegl, Linear Spaces and Differentiation Theory, J. Wiley and Sons, New York, 1988. [8] M.W. Hirsch, Differential Topology, GTM 33, Springer-Verlag, New York, 1976. [9] I.M. James, General Topology and Homotopy Theory, Springer-Verlag, Berlin, 1984. [10] A. Kriegl, P. Michor, Convenient Settings of Global Analysis, Am. Math. Soc., 1997. [11] Jet Nestruev, Smooth Manifolds and Observables, Springer-Verlag New York, Inc., 2003 [12] G.W. Whitehead, Elements of Homotopy Theory, Springer-Verlag, New York, 1978. 27