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f from ,t x, etlpee n nodte noa into them unfold and pieces mental oteretraction the to optr n opsn hmby them composing and computer, a ) inlymr ffiin sats for task a is efficient more tionally fcmue oeig hc has which modeling, computer of f ∈ [ y nweg h rtmto was method first the knowledge e hnapofb formula. a by proof a than ter ◦ slspeetdb o 1 and [1] Fox by presented esults ] X g X M e × Y to e f g ntrsof terms in htis, That . I Let . : 1 and , f rasrn deformation strong a or Y Y noistop. its onto . X r e y : = X x, ∈ M hnw have we Then . q Y 0) f f r ( , X X respectively. , e sdfie by defined is ∼ g ×{ , sastrong a is f F M ( 1 x f and } .The ). ,and ), given G . (2) Use f and G to define a map H2 : Mf × I Mf by

[x, t],s [G(f(x), 1 − s)] and [y],s [G(y, 1 − s)]

This is a well defined homotopy from r to a map h : Mf Mf defined by [x, t] [g ◦ f(x), 0] and [y] [g(y), 0].

(3) Use g and F to define a map H3 : Mf × I Mf by

[x, t],s [F (x,st),s] and [y],s [g(y),s]

′ This is a well defined homotopy from the map h to a retraction r : Mf Mf of the mapping cylinder onto its top X defined by

e [x, t] [F (x, t), 1] and [y] [g(y), 1].

′ The concatenation H = H1 ∗ H2 ∗ H3 is the desired deformation retraction from 1Mf to a retraction r of Mf onto its top.

2.2. The Homotopy extension property of (Mf × I, X × I) made explicit: We need to construct a R M I I M I I M I I M I X I I retraction : f × × f × × of f × × ontof f × ×{0} ∪ × × . This is achieved by defining first a retraction ϕ : I2 I2 of I2 onto I ×{0}∪{1}× I via radial projection from the point e (0, 2), as the figure below illustrates.

v 2u , 0 if v ≤ 2 − 2u (0, 2) 2 − v  ϕ(u, v)=    2u + v − 2 1, if v ≥ 2 − 2u.  u   

If we let ϕ1 and ϕ2 denote the components of ϕ the retraction R is defined by

[x, t],s,l [x, ϕ1(t,l)],s,ϕ2(t,l) and [y],s,l [y], s, 0 . u

3. Construction

In order to facilitate clarity a point in Mf will be denoted by the letter p. If the point is in X (i.e. p = [x, 1]) we will denote it byp ˜. e Our goal is to modify the deformation retraction H constructed earlier so that it leaves X fixed. We begin by defining a homotopy K : M × I M by f f e 1 H−1(p, 1 − 2s) if 0 ≤ s ≤  2 K(p,s)=   1 H−1(r′(p), 2s − 1) if ≤ s ≤ 1, 2   where H−1 denotes the inverse of H (i.e. H with s running backward). Roughly speaking, the first part of K is simply H−1 twice as fast, while the second half is a map that begins and ends with r′, and is homotopic1 −1 ′ to H . It easy to check that K is a well defined homotopy from 1Mf to r , and that its restriction to X × I is a homotopy beginning and ending with 1 e . Moreover, it behaves well with respect to time reversal in the X e sense that K(˜p, 1 − s)= K(˜p,s).

1 1 ′ 1 1 ′ 1 −1 −1 ′ 1 Because Mf ∼ r implies that Mf × I ∼ r × I , and therefore H ∼ H ◦ (r × I ). 2 The next step will be to define a homotopy L : (X × I) × I Mf . We use K and the diagram below to accomplish this. e Roughly speaking, we want L(˜p,s,u) to be independent of u below the ‘V’, and u independent of s above the ‘V’. More formally:

K(˜p,s) if u ≤ |2s − 1|  L(˜p,s,u)=  1 − u K p,˜ if 2s − u ≤ 1 ≤ 2s + u. s  2  1 ′  Mf r 

2 This is a well defined homotopy from K Xe×I to a map sending (˜p,s) top ˜. Moreover, L(˜p,s,u)=p ˜ for

(s,u) ∈ ∂I × I ∪ I ×{1}. Therefore, looking at the diagram above we would like to extend L to the whole mapping cylinder and then follow this extension from the lower left corner 1Mf along the left, top and right ′ edges to the right corner r . Indeed, define a map K0 : Mf × I ×{0} Mf by K0(p, s, 0) = K(p,s), and 3 ′ combine it with L to give a map (K0,L): Mf × I ×{0} ∪ X × I × I Mf . Then, define L by the composition R e (K0,L) Mf × I × IMf × I ×{0} ∪ X × I × IMf , ′ This map clearly extends L and satisfies L (p, s, 0) = K(p,s).e Finally, define a map Γ : Mf × I Mf by

1 L′(p, 0, 3s) if 0 ≤ s ≤  3    ′ 1 2 Γ(p,s)= L (p, 3s − 1, 1) if ≤ s ≤  3 3

 ′ 2 L (p, 1, 3 − 3s) if ≤ s ≤ 1.  3   It is easy to check that this map verifies the following: • Γ(p, 0) = L′(p, 0, 0) = K(p, 0) = p, • Γ(p, 1) = L′(p, 1, 0) = K(p, 1) = r′(p), • Γ(˜p,s)=˜p because L′ is an extension of L. ′ Therefore, Γ is the desired strong deformation retraction from the identity 1Mf to a retraction r of the mapping cylinder onto its top X.

e 4. Formula If we unravel Γ we will obtain the desired formula in terms of f, g, F and G. It is given in the next two pages.

2 1−u 1+u Because K(˜p, 1 − s)= K(˜p, s) implies that K(˜p, 2 )= K(˜p, 2 ). 3 e This map is unambiguously defined because K0 = L on the overlap X × I × {0}. Continuity follows from the gluing lemma e because X is a closed subset of in Mf . 3 2t 2 − 2t x, if s ≤  1  2 − 3s 3 if 0 ≤ s ≤   3    2 − 2t  [x, 1] if s ≥  3          1 7  x, t s s  [ 2 (7 − 18 )] if ≤ ≤    3 18       7 8   [G(f(x), 8 − 18s)] if ≤ s ≤    18 18          8 1   [F (x, 2t(18s − 8)) , 18s − 8] if ≤ s ≤   1  18 2  if t ≤       2  1 10   [F (F (x, 2t), 10 − 18s) , 10 − 18s] if ≤ s ≤   2 18          10 11   [G (f ◦ F (x, 2t), 18s − 10)] if ≤ s ≤    18 18          11 2   [F (x, 2t), 18s − 11] if ≤ s ≤    18 3     1 2   if ≤ s ≤  Γ [x, t],s =  3 3   1 7
x, s s  [ 7 − 18 ] if ≤ ≤    3 18      7 8      [G(f(x), 8 − 18s)] if ≤ s ≤    18 18       8 1   [F (x, 18s − 8) , 18s − 8] if ≤ s ≤      1  18 2  if t ≥    2  1 10   [F (x, 10 − 18s) , 10 − 18s] if ≤ s ≤     2 18       10 11      [G(f(x), 18s − 10)] if ≤ s ≤    18 18       11 2   [x, 18s − 11] if ≤ s ≤       18 3             2t 1+2t  F x, , 1 if s ≥    3s − 1  3  2  if ≤ s ≤ 1   3    1+2t  [x, 1] if s ≤  3    

4 7 [y] if 0 ≤ s ≤  18   7 8 [G(y, 8 − 18s)] if ≤ s ≤  18 18   8 1  g y , s s [ ( ) 18 − 8] if ≤ ≤  18 2   1 10 Γ [y],s = [F (g(y), 10 − 18s)] if ≤ s ≤  2 18
  10 11 [G (f ◦ g(y), 18s − 10)] if ≤ s ≤  18 18   11 2 [g(y), 18s − 11] if ≤ s ≤   18 3   2 [g(y), 1] if ≤ s ≤ 1  3  

References

[1] R.H. Fox, On homotopy type and deformation retracts, Ann. of Math. 44, no. 1 (1943), 40-50. [2] M. Fuchs, A note on mapping cylinders, Michigan Math. J. 18, no. 4 (1971), 289-290. [3] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.

Department of Mathematics, Duke University, Durham, NC 27708-0320 E-mail address: [email protected]

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