# Tagged Questions

Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

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### Let ($\mathbb{R}^n$,*) be a division algebra over $\mathbb{R}$, $n>1$, $\Rightarrow n$ is even

I want to prove: Let ($\mathbb{R}^n$,*) be a division algebra over $\mathbb{R}$, $n>1$, $\Rightarrow n$ is even. I'm stuck. My thoughts are: I want to use the hairy ball theorem and I want to ...
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### Alexander Method for non-orientable Surface

From Farb and Margalit,Primer on Mapping Class Group, p.62, we know For example for torus with four puncture we can choose following claret red curves. Curves are choosen such that their complement ...
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### Milnor's proof of the fundamental theorem of algebra (Topology from the Differentiable Viewpoint)

I am studying the proof of the fundamental theorem of algebra out of John Milnor's book Topology from the Differentiable Viewpoint, located on page 8 here: http://webmath2.unito.it/paginepersonali/...
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### Transversality through two functions $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ for $W\subset Z$

For Exercise 5 section 5 chapter 2 of Guillemin & Pollak: Set $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ and assume that g is transversal to a submanifold $W\subset Z$. Show $f\pitchfork g^{-1}(W)$ ...
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### Why the map $z→z+ \overline z^m$ has fixed point with local Lefschetz number $m$ at the the origin of C (m≥0)?

My professor went through an example in class and making following claim The map $z→z+z^m$ has a fixed point with local Letfschetz number $m$ at the origin of $C$ $(m>0)$ For any $c≠0$, the ...
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### Only one fixed point for $f:\bar{\mathbb{D}}\rightarrow\bar{\mathbb{D}}$ on the boundary.

We know for Brouwer theorem that $f$ (continuous bijective function) have a fixed point. My questions are: 1) Is there a function with only one fixed point $x_0\in Int(\bar{\mathbb{D}})$ (open disk)?...
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### 1-forms as a direct sum

Let us define the spaces $C_0, C_1$ and $C_2$ of differential $0,1,2$ forms respectively on the sphere $S^2.$ Is it true that $C_1$ is the direct sum of $d(C_0) \oplus \delta^* (C_2)$? I think this ...
Assume $K$, $L$ are $n$-pseudomanifold, and $K$ is compact. Let $f$ be a simplicial map between $K$ and $L$. We denote $n$-simplexes of $K$ and $L$ by $S_n(K)$, $S_n(L)$. Define \deg_2f:=\text{#}\{\...