Degree one map: Difference between revisions

Latest revision as of 19:43, 11 May 2008

This article defines a property of a continuous map between compact connected orientable manifolds (the property may strictly require a choice of orientation)

Definition

Let $M$ and $N$ be compact connected orientable manifolds. Choose fundamental classes for $M$ and $N$ . Then a degree one map from $M$ to $N$ is a continuous map $f : M \to N$ such that $f$ sends the fundamental class of $M$ to the fundamental class of $N$ .

Facts

There always exists a degree one map from any compact connected orientable manifold, to the sphere. The map is constructed as follows: let $M$ be the compact connected orientable manifold, and $p \in M$ be a point. Suppose $D$ is a closed disc containing $p$ inside a Euclidean neighbourhood of $p$ . Let $A$ denote the complement of $D$ . The map $M \to M / A = S^{n}$ is a degree one map, viz it induces an isomorphism on the $n^{t h}$ homology.
In general, if $M$ and $N$ are compact connected orientable manifolds, then there exist degree one maps from $M ♯ N$ to $M$ and to $N$ . The map to $M$ , for instance, pinches the entire part from $N$ to a point. The previous fact can be viewed as a special case of this, by viewing $M$ as a connected sum $M ♯ S^{n}$ .
Any degree one map induces a surjective map on fundamental groups, and hence in particular on the first homology group
Any degree one map induces a surjective map on all homology groups

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 ==Definition==
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 * There always exists a degree one map from any compact connected orientable manifold, to the sphere. The map is constructed as follows: let <math>M</math> be the [[compact connected orientable manifold]], and <math>p \in M</math> be a point. Suppose <math>D</math> is a closed disc containing <math>p</math> inside a Euclidean neighbourhood of <math>p</math>. Let <math>A</math> denote the complement of <math>D</math>. The map <math>M \to M/A = S^n</math> is a degree one map, viz it induces an isomorphism on the <math>n^{th}</math> homology.
-* In general, if <math>M</math> and <math>N</math> are [[compact connected orientable manifold]]s, then there exist degree one maps from <math>M \sharp N</math> to <math>M</math> and to <math>N</math>. The map to <math>M</math>, for instance, pinches the entire part from <math>N</math> to a point.
+* In general, if <math>M</math> and <math>N</math> are [[compact connected orientable manifold]]s, then there exist degree one maps from <math>M \sharp N</math> to <math>M</math> and to <math>N</math>. The map to <math>M</math>, for instance, pinches the entire part from <math>N</math> to a point. The previous fact can be viewed as a special case of this, by viewing <math>M</math> as a connected sum <math>M \sharp S^n</math>.
+* Any degree one map induces a surjective map on [[fundamental group]]s, and hence in particular on the [[first homology group]]
+* Any degree one map induces a surjective map on all homology groups