Deformation retraction

cjerdonek authored on Nov. 27, 2020, 12:50 a.m.

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In mathematics (and topology in particular), a deformation retraction captures the idea of deforming a space over time into a subspace of itself. This figure illustrates an example of a deformation retraction of a disk into a subspace consisting of a single point. Specifically, given a topological space \(X\) and a subspace \(A\) of \(X\), a deformation retraction from \(X\) to \(A\) is a homotopy from the identity map on \(X\) to a map from \(X\) to \(A\) that is the identity when restricted to \(A\). Examples For example, the first figure above illustrates a deformation retraction from a disk to one of its points. An annulus does not admit a deformation retraction to a point, like the one shown in bold. However, it does admit a deformation retraction to a core curve (also shown in bold). In contrast, an annulus (like the one pictured on the right) does not admit a deformation retraction to a point. One way to see this is that a core curve, like the one in the interior shown in bold, is homotopically nontrivial, while a deformation retraction to a point would imply that the curve is homotopically trivial. An annulus, however, can easily be seen to admit a deformation retraction to a core curve. Strong deformation retraction If the homotopy in the definition of a deformation retraction is the identity on \(A\) for all maps in the family (so not just the first and final maps), then the deformation retraction is said to be a strong deformation retraction. The figure below illustrates the difference between a deformation retraction that is strong and one that isn't. This figure shows what it means for a deformation retraction to be strong versus not strong. Both the left and right show a deformation retraction from a square disk to the subspace defined by its bottom edge. The retraction on the left is strong. However, the one on the right isn't strong because the edge subspace doesn't map to itself throughout the entire deformation.	In mathematics (and topology in particular), a deformation retraction captures the idea of deforming a space over time into a subspace of itself. This figure illustrates an example of a deformation retraction of a disk into a subspace consisting of a single point. Specifically, given a topological space \(X\) and a subspace \(A\) of \(X\), a deformation retraction from \(X\) to \(A\) is a homotopy from the identity map on \(X\) to a map from \(X\) to \(A\) that is the identity when restricted to \(A\). Examples For example, the first figure above illustrates a deformation retraction from a disk to one of its points. An annulus does not admit a deformation retraction to a point, like the one shown in bold. However, it does admit a deformation retraction to a core curve (also shown in bold). In contrast, an annulus (like the one pictured on the right) does not admit a deformation retraction to a point. One way to see this is that a core curve, like the one in the interior shown in bold, is homotopically nontrivial, while a deformation retraction to a point would imply that the curve is homotopically trivial. An annulus, however, can easily be seen to admit a deformation retraction to a core curve. Strong deformation retraction If the homotopy in the definition of a deformation retraction is the identity on \(A\) for all maps in the family (so not just the first and final maps), then the deformation retraction is said to be a strong deformation retraction. Examples The figure below shows an example illustrating the difference between a deformation retraction that's strong and one that isn't. The pictures on the left and right both show a deformation retraction from the square disk to the subspace defined by its bottom edge. The one on the left is strong because the homotopy preserves the subspace for all times. The one on the right isn't, however, because the edge subspace doesn't map to itself throughout the entire deformation. This figure shows what it means for a deformation retraction to be strong versus not strong. The left and right both depict deformation retractions from the square disk to its bottom edge. The one on the left is strong, while the one on the right isn't.

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In mathematics (and topology in particular), a deformation retraction captures the idea of deforming a space over time into a subspace of itself.

This figure illustrates an example of a deformation retraction of a disk into a subspace consisting of a single point.

Specifically, given a topological space \(X\) and a subspace \(A\) of \(X\), a deformation retraction from \(X\) to \(A\) is a homotopy from the identity map on \(X\) to a map from \(X\) to \(A\) that is the identity when restricted to \(A\).

Examples

For example, the first figure above illustrates a deformation retraction from a disk to one of its points.

An annulus does not admit a deformation retraction to a point, like the one shown in bold. However, it does admit a deformation retraction to a core curve (also shown in bold).

In contrast, an annulus (like the one pictured on the right) does not admit a deformation retraction to a point. One way to see this is that a core curve, like the one in the interior shown in bold, is homotopically nontrivial, while a deformation retraction to a point would imply that the curve is homotopically trivial. An annulus, however, can easily be seen to admit a deformation retraction to a core curve.

Strong deformation retraction

If the homotopy in the definition of a deformation retraction is the identity on \(A\) for all maps in the family (so not just the first and final maps), then the deformation retraction is said to be a strong deformation retraction. The figure below illustrates the difference between a deformation retraction that is strong and one that isn't.

This figure shows what it means for a deformation retraction to be strong versus not strong. Both the left and right show a deformation retraction from a square disk to the subspace defined by its bottom edge. The retraction on the left is strong. However, the one on the right isn't strong because the edge subspace doesn't map to itself throughout the entire deformation.

In mathematics (and topology in particular), a deformation retraction captures the idea of deforming a space over time into a subspace of itself.

Examples

For example, the first figure above illustrates a deformation retraction from a disk to one of its points.

Strong deformation retraction

Examples

The figure below shows an example illustrating the difference between a deformation retraction that's strong and one that isn't. The pictures on the left and right both show a deformation retraction from the square disk to the subspace defined by its bottom edge. The one on the left is strong because the homotopy preserves the subspace for all times. The one on the right isn't, however, because the edge subspace doesn't map to itself throughout the entire deformation.

This figure shows what it means for a deformation retraction to be strong versus not strong. The left and right both depict deformation retractions from the square disk to its bottom edge. The one on the left is strong, while the one on the right isn't.

cjerdonek / Deformation retraction 08174c3afe923debf3790e026bca0a50b36b9d85

Examples

Strong deformation retraction

Examples

Strong deformation retraction

Examples