Mixed structure of approximate Serre co-fibration

Introduction

In this paper there is a description of the mixed approximate Serre co-fibration and approximate homotopy extension property. In [1,2,3] mention if 𝑞: 𝐸 → 𝐵 is a function has an approximate lowering homotopy property (ALHP) with respect to 𝑋 provided that given ξ is open cover of 𝑋 if and only if define a function 𝑓: 𝐵 → 𝑌 and a homotopy 𝑔 𝑡: 𝐸 → 𝑌 satisfying 𝑓 ∘ 𝑞 = 𝑔 0 , then there exists a homotopy 𝑓 𝑡: 𝐵 → 𝑌 with 𝑓 0 = 𝑓 and 𝑓 𝑡 ∘ 𝑞 is ξclosed 𝑔 𝑡 for all 𝑡 ∈ 𝐼.

Also [1,4] The map 𝑞 is co-fibration if it has the approximate lowering homotopy property. In addition to that [1,5] assume 𝑞: 𝐸 → 𝐵 be a continuous function of spaces, 𝑞 has the approximate lowering homotopy property (ALHP) with respect to a CW-complex space 𝑌.

Diagram 1: (L.H.P)

If 𝑌 is the space of all 𝐶𝑊complex space or (all finitely triangulable spaces), then 𝑞: 𝐸 → 𝐵 is called an approximate Serre cofibration.

Mixed approximate Serre co-fibration

Let 𝑓 1: 𝐸 1 → 𝑍, 𝑓 2: 𝐸 2 → 𝑍 are two fiber space and 𝛼: 𝐸 2 → 𝐸 1 where 𝑓 1 ∘ 𝛼 = 𝑓 2 , 𝑙𝑒𝑡 𝐸 𝑖 = {𝐸 1 , 𝐸 2 }, 𝑓 𝑖 = {𝑓 1 , 𝑓 2 } with Z be a CWcomplex the {𝐸 𝑖 , 𝑓 𝑖 , 𝑍, 𝛼}, has Mixed approximate lowering homotopy property (MALHP) [2] with respect to a CW-space 𝑌 iff given a map 𝑘: 𝑍 → 𝑌 and a homotopy 𝑔 𝑡: 𝐸 1 → 𝑌 satisfying 𝑘 ∘ 𝑓 2 = 𝑔 0 ∘ 𝛼 then there exist a homotopy 𝑘 𝑡: 𝑍 → 𝑌 with 𝑘 0 = 𝑘 and 𝑘 𝑡 ∘ 𝑓 1 = 𝑔 𝑡 for all 𝑡 ∈ 𝐼. M-fiber space is called M approximate Serre co-fibration. 𝜆 1 (𝑣 1 ) = 𝑓 1 (𝑣 1 ) = ℎ 1 (𝑞 1 (𝑣 1 ))

Homotopy Extension Property

Since 𝑞 1 (𝑢 1 ) = 𝑞 1 (𝑣 1 ), therefore 𝜆 1 (𝑢 1 ) = 𝜆 1 (𝑣 1 ) and

Therefore 𝑞 𝑖 must be injective. Since 𝐸 𝑖 is compact Hausdorff and 𝐵 is Hausdorff, hence 𝑞 𝑖 must be an imbedding.

Hence 𝑞 1: 𝐸 2) A continuous function 𝜑 𝑖: 𝑌 𝑖 → 𝐼 such that 𝐴 1 = 𝜑 1 -1 (0), 𝜑 1 (𝑦 1 ) = 1 and

Proof: Suppose that (𝑌 𝑖 , 𝐴 𝑖 ) is a M-Serre cofibered pair. Then there exists a retraction

Where 𝑖 = 1,2, and 𝑈 𝑖 , 𝐻 𝑖 and 𝜑 𝑖 may be chosen as follows:

𝑝𝑟 1 and 𝑝𝑟 2 denoting projections on 𝑌 𝑖 and 𝐼, respectively. Conversely, suppose that 𝑈 𝑖 , 𝐻 𝑖 and 𝜑 𝑖 are given and satisfy the conditions of the theorem. Since 𝐴 𝑖 is closed it suffices to prove the existence of a retraction.

𝑟 𝑖: 𝑌 𝑖 × 𝐼 → (𝑌 𝑖 × 0) ∪ (𝐴 𝑖 × 𝐼)

The required retraction may be constructed as follows: a) If 𝜑 𝑖 (𝑦 𝑖 ) = 1, let 𝑟 𝑖 (𝑦 𝑖 , 𝑡) = (𝑦 𝑖 , 0). b) If

and 2𝜑 𝑖 (𝑦 𝑖 ) ≤ 𝑡 ≤ 1, let 𝑟 𝑖 (𝑦 𝑖 , 𝑡) = {(𝐻 1 (𝑦 1 , 1), 𝑡 -2𝜑 1 (𝑦 1 )). (𝐻 2 (𝑦 2 , 1), 𝑡 -2𝜑 2 (𝑦 2 )). e) If 𝜑 𝑖 (𝑦 𝑖 ) = 0, let 𝑟 𝑖 (𝑦 𝑖 , 𝑡) = (𝑦 𝑖 , 𝑡). Theorem 3.6. Suppose that 𝑝 𝑖: 𝐸 𝑖 → 𝐵 is M-fibration, that 𝐴 𝑖 is a strong deformation retract of 𝑌 𝑖 , and that there exists a map 𝜑 𝑖: 𝑌 𝑖 → 𝐼 such that 𝐴 1 = 𝜑 1 -1 (0)and 𝐴 2 = 𝜑 2 -1 (0).Then any commutative diagram may be filled in with a map ℎ 𝑖: 𝑌 𝑖 → 𝐸 𝑖 such that 𝑝 1 ℎ 1 = 𝑓 1 , 𝑝 2 ℎ 2 = 𝑓 2 and ℎ 1 𝜂 1 = 𝑓 1 ́, ℎ 2 𝜂 2 = 𝑓 2 ́. ℎ 𝑖 is unique up to homotopy 𝑟𝑒𝑙𝐴 𝑖 .

Proof: By hypothesis there exists a retraction 𝑟 1: 𝑌 1 → 𝐴 1 ,𝑟 2: 𝑌 2 → 𝐴 2 and a homotopy

If ℎ 1: 𝑌 1 → 𝐸 1 and ℎ 2: 𝑌 2 → 𝐸 2 such that ℎ 𝑖 𝜂 𝑖 = 𝑓 𝑖 ́, then ℎ 𝑖 ≃ ℎ 𝑖 𝜂 𝑖 𝑟 𝑖 = 𝑓 𝑖 ́ 𝑟𝑒𝑙𝐴 which proves the last assertion of the theorem. Define 𝐷 𝑖: 𝑌 𝑖 × 𝐼 → 𝑌 𝑖 by

𝐷 𝑖 is easily shown to be continuous. Because p i is a fibration there exists a homotopy 𝐹 1: 𝑌 1 × 𝐼 → 𝐸 1 , 𝐹 2: 𝑌 2 × 𝐼 → 𝐸 2 such that 𝑝 1 𝐹 1 = 𝑓 1 𝐷 1 , 𝑝 1 𝐹 1 = 𝑓 1 𝐷 1 and 𝐹 1 (𝑦 1 , 0) = 𝑓 1 ́𝑟1 (𝑦 1 ), 𝐹 2 (𝑦 2 , 0) = 𝑓 2 ́𝑟2 (𝑦 2 ) for each 𝑦 1 ∈ 𝑌 1 , 𝑦 2 ∈ 𝑌 2 . ℎ 𝑖 is given by ℎ 𝑖 (𝑦 𝑖 ) = 𝐹 𝑖 (𝑦 𝑖 , 𝜑 𝑖 (𝑦 𝑖 )), where 𝑖 = 1,2.

Theorem 3.7. Suppose that 𝑝 𝑖: 𝐸 𝑖 → 𝐵 is a mixed approximate fibration, that (𝑌 𝑖 , 𝐴 𝑖 ) is a mixed approximate Serre cofibered pair, and that 𝐴 𝑖 is closed. Then any commutative diagram may be filled in with a homotopy 𝐹: 𝑌 𝑖 × 𝐼 → 𝐸 𝑖 such that 𝑝 1 𝐹 1 = 𝐹 1 , 𝑝 2 𝐹 2 = 𝐹 2 and 𝐹 𝑖 |(𝑌 𝑖 × 0) ∪ (𝐴 𝑖 × 𝐼) = 𝑓 𝑖 .

Proof: According to the lemma(4.5), and by theorem(4.4) there exists a Function 𝜓 1: 𝑌 1 → 𝐼, 𝜓 2: 𝑌 2 → 𝐼 such that 𝐴 1 = 𝜓 1 -1 (0), 𝐴 2 = 𝜓 2 -1 (0).

Define 𝜑 1: 𝑌 1 × 𝐼 → 𝐼 and 𝜑 2: 𝑌 2 × 𝐼 → 𝐼 by 𝜑 1 (𝑦 1 , 𝑡) = 𝑡𝜓 1 (𝑦 1 )and 𝜑 2 (𝑦 2 , 𝑡) = 𝑡𝜓 2 (𝑦 2 ). Then (𝑌 1 × 0) ∪ (𝐴 1 × 𝐼) = 𝜑 1 -1 (0)and (𝑌 2 × 0) ∪ (𝐴 2 × 𝐼) = 𝜑 2 -1 (0) and the theorem follows from theorem(4.6) The condition that 𝐴 𝑖 be closed is not very restrictive. For instance, A will always be closed if 𝑌 𝑖 is Hausdorff. Not all M-Serre cofibration are closed. The most trivial example of a non-closed M-Serre cofibration is the pair (𝑌 1 , 𝑎 1 ), (𝑌 2 , 𝑎 2 ) where 𝑌 𝑖 is the two-point space 𝑎 𝑖 , 𝑏 𝑖 with the trivial topology 4. A mixed criterion for a map to be a mixed APPROXIMATE Serre Co-fibration The M-criterion, which enables us to identify M-Serre Cofibration when we observe it, is discussed in this section. We frequently examine pairs (𝑌 𝑖 , 𝐴 𝑖 ) that comprise a space 𝐴 𝑖 . and a subspace 𝐴 𝑖 . Pairs that "behave homologically" in the same way as the corresponding quotient spaces 𝑌 𝑖 𝐴 𝑖 . are known as M-Serre Cofibration pairs. Definition 4.1. A pair (𝑌 𝑖 , 𝐴 𝑖 ) is an Mixed Neighborhood Deformation Retract pair (MNDR-pair) if there is a map 𝑢: 𝑌 1 → 𝐼, 𝑣: 𝑌 2 → 𝐼 such that 𝑢 -1 (0) = 𝐴 1 , 𝑣 -1 (0) = 𝐴 2 and a homotopy ℎ: 𝑌 1 × 𝐼 → 𝑌 1 , 𝑘: 𝑌 2 × 𝐼 → 𝑌 2 such that ℎ 0 = 𝑖𝑑, ℎ(𝑎 1 , 𝑡) = 𝑎 1 and 𝑘 0 = 𝑖𝑑, 𝑘(𝑎 2 , 𝑡) = 𝑎 2 for 𝑎 1 ∈ 𝐴 Where 𝑘 1 (𝑎 1 ) = (𝑎 1 , 1) and 𝑘 2 (𝑎 2 ) = (𝑎 2 , 1) where 𝑀 𝑖 𝜂 𝑖 ≡ 𝑌 𝑖 ∪ 𝜂 (𝐴 𝑖 × 𝐼). The obviou homotopy inverse 𝑙 𝑖: 𝑌 𝑖 → 𝑀 𝑖 𝜂 𝑖 has 𝑙 𝑖 (𝑦 𝑖 ) = (𝑦 𝑖 , 0) and is thus very far from being a map 𝐴 𝑖 . The proposition guarantees that l i is homotopic to a map under 𝐴 𝑖 that is homotopy inverse to 𝑟 𝑖 under 𝐴 𝑖

Conclusion

We mention the most important result that reached through our study mixed structure Serre cofibration as following:

• Every approximate Serre cofibration is Mixed approximate Serre cofibration.

• product Two Mixed approximate Serre cofibration is also Mixed approximate Serre cofibration.

• M-pullback of Mixed approximate Serre cofibration is also Mixed approximate Serre cofibration.

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