# How can I see that this map is a null homotopy?

Let $$A\subset\mathbb R^n$$ be star-convex, that is there exists $$a_0\in A$$ such that for each $$a\in A$$, the line segment from $$a_0$$ to $$a$$ is contained in $$A$$. Define $$H:A\times[0,1]\to\mathbb R^n$$ by $$H(a,t)=(1-t)a + ta_0$$.

The book (Dieck's Algebraic Topology) claims that $$H$$ is a null homotopy of the identity, and hence star-convex sets are contractible.

I see that a null homotopy of a map $$f$$ is a homotopy between $$f$$ and a constant map, and that a null homotopy of the identity is a contraction of the underlying space. However, I do not see how the homotopy $$H$$ defined here is a null homotopy, or that it is a null homotopy of the identity. Can anyone help me to understand this?

## 1 Answer

Note that $$H(a,t)$$ is obviously continuous: addition and scalar multiplication of continuous functions $$\mathbb R^n\to \mathbb R^n$$ yields new continuous functions from old. For each $$a \in \mathbb R^n$$, $$H(a,0) = (1-0)a + 0\cdot a_0 = a$$. On the other hand, $$H(a,1) = (1-1)a + 1\cdot a_0 = a_0$$. Therefore $$H(\cdot,0)$$ is the identity map, while $$H(\cdot, 1)$$ is a constant map at $$a_0$$. This demonstrates that the map is a contraction.

• Where does the "null homotopy" come into play exactly? The fact that $H(\cdot,0)$ is the identity map? Would I be correct in saying that $H$ is a homotopy between the identity map and the constant map $a_0\mapsto a_0$? I am just beginning to study these concepts and it is very confusing to me. Dec 7, 2019 at 1:32
• Note that the constant map is emphatically not just $a_0 \mapsto a_0$ but $a \mapsto a_0$ for all $a \in A$. The word “null” in null homotopy comes from thinking of constant maps as somehow “zero”. It is not more complicated than that. Dec 7, 2019 at 1:35
• It's just terminology: A function $f$ is null-homotopic if it is homotopic to a constant function. The homotopy itself is what is called the null-homotopy. Dec 7, 2019 at 1:38
• @RyleeLyman Yes, $a\mapsto a_0$ for all $a\in A$ is what I meant. And I can see how "null" would be associated with constant maps. Thank you. Dec 7, 2019 at 1:40
• To make sure I understand though: $H$ is a homotopy between the identity map on $A$ and the constant map $a\mapsto a_0$ on $A$, correct? Dec 7, 2019 at 1:43