Problem 1 - Algebraic Topology

Question

Given maps \(f_{i}: X_{i} \rightarrow X_{i+1}\) for integers \(i<0,\) show that the 'reverse mapping telescope' obtained by glueing together the mapping cylinders of the \(f_{i}\) 's in the obvious way deformation retracts onto \(X_{0} .\) Similarly, if maps \(f_{i}: X_{i} \rightarrow X_{i+1}\) are given for all \(i \in \mathbb{Z},\) show that the resulting 'double mapping telescope' deformation retracts onto any of the ordinary mapping telescopes contained in it, the union of the mapping cylinders of the \(f_{i}\) 's for \(i\) greater than a given number \(n\).

Step-by-Step Solution

Answer

The reverse mapping telescope can be retracted onto \( X_0 \), and the double mapping telescope retracts onto any part past a chosen \( n \).

Step 1: Understand the Mapping Cylinder Construction

A mapping cylinder of a map \( f_i: X_i \rightarrow X_{i+1} \) can be understood as the topological space obtained by taking the product \( X_i \times [0,1] \) and gluing one end along the mapping \( f_i \). This forms a sort of 'tube' that connects the space \( X_i \) to \( X_{i+1} \).

Step 2: Build the Reverse Mapping Telescope

To construct the reverse mapping telescope for \( i < 0 \), we 'glue' these mapping cylinders \( M(f_i) \) end-to-end for the sequence of maps \( \ldots \rightarrow X_{-2} \rightarrow X_{-1} \rightarrow X_0 \), forming an 'infinite tube-like' space moving backwards from \( X_0 \).

Step 3: Deformation Retraction Onto \( X_0 \)

A deformation retraction from the entire reverse telescope onto \( X_0 \) means we can continuously shrink the complex, pulling it entirely onto \( X_0 \) without cutting or tearing. This is done by using the continuity of the mappings to 'slide' along the cylinders, retracting to the base space \( X_0 \) as all preceding stages of mapping ultimately funnel down into \( X_0 \).

Step 4: Construct the Double Mapping Telescope

For maps \( f_i: X_i \rightarrow X_{i+1} \) given for all \( i \in \mathbb{Z} \), we build on both forward and backward mappings. This creates a bi-infinite sequence of mapping cylinders forming a 'double telescope' connecting through all integers across the spectrum of \( \ldots, X_{-2}, X_{-1}, X_0, X_1, X_2, \ldots \).

Step 5: Deformation Retraction of the Double Telescope

In the double mapping telescope, pick a number \( n \) and consider the ordinary telescope for \( i > n \). The entire double telescope will deformation retract onto any part of itself because each portion \( X_n \) acts as a base stabilizing point with self-sufficient funneling properties, showing structural unity across the telescopes. Continuity of mappings ensures we can contract the structure including any chosen part, effectively onto \( X_n \).

Key Concepts

Mapping Cylinder

The concept of a mapping cylinder is intriguing in the world of topology. Imagine you have a map, like a function, that connects two spaces, say from space \( X_i \) to \( X_{i+1} \). The mapping cylinder, denoted as \( M(f_i) \), is created by taking the product \(X_i \times [0,1]\). Here, \([0,1]\) represents the unit interval, a continuous stretch of numbers between 0 and 1.

This product forms a sort of tube which you then modify by attaching one end to \(X_{i+1}\) via the map \(f_i\). Think of it like a corridor stretching between two spaces, but its ends are connected in a way that reflects the properties of the map. This forms a smooth transition or "glue" between \(X_i\) and \(X_{i+1}\), effectively creating a continuous morphing from one space to another.

• It's a way of visualizing maps as actual shapes in space.
• The cylinder helps to "smooth out" the transition between mapped spaces.
• This construction is fundamental in creating mapping telescopes, where each mapping cylinder is a segment.

Double Mapping Telescope

Imagine you have a sequence of spaces connected by maps both forward and backward, like a bi-directional chain. This is the foundation of a double mapping telescope. It involves connecting an endless sequence of mapping cylinders from the set of maps \(f_i: X_i \to X_{i+1}\) for each integer \(i\).

In this construction, you are building endless "corridors" forward in time \((i \to i+1, i+1 \to i+2, \ldots)\) and backward \((i \to i-1, i-1 \to i-2, \ldots)\), creating a vast network that moves across positive and negative infinity. This double connection represents a mapping "tunnel," uniting all referenced spaces into a graceful, infinite corridor.

• This telescope features a consistent tubular structure from negative to positive infinity.
• Think of it like constructing interconnecting tunnels, with each map providing seamless connectivity.
• The double nature ensures you can focus on any specific part \(i>n\), and it will still retract inward to any portion or base part effortlessly.

Deformation Retraction

Deformation retraction is a fantastic concept in topology. It's like being able to continuously fold a vast, complex surface into a simpler sub-space without tearing it apart. When we say a space deformation retracts onto a subspace like \(X_0\), it means you can smoothly pull and contract the entirety of that space until it perfectly fits onto \(X_0\).

In the reverse mapping telescope, deformation retraction allows the sequence of mapping cylinders to shrink back to the base space\(X_0\). This process involves making each point in the telescope "slide" along the mapping cylinders, moving smoothly downwards into \(X_0\).

• Allows large complex structures to become simpler and more manageable.
• Highlights the continuous, fluid nature of the topological transformations.
• In the telescope structures, it's akin to watching a massive structure funnel naturally through to the original \(X_0\) into which everything eventually merges.