# 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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### What is the relationship between diffeomorphisms of the sphere modulo isotopy and exotic spheres?

In his "Classification of (n-1)-connected 2n-dimensional manifolds and the discovery of exotic spheres", Milnor observes that since his exotic 7-spheres admit a Morse function with only two critical ...
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### Orientability of $m\times n$ matrices with rank $r$

I know that $$M_{m,n,r} = \{ A \in {\rm Mat}(m \times n,\Bbb R) \mid {\rm rank}(A)= r\}$$is a submanifold of $\Bbb R^{mn}$ of codimension $(m-r)(n-r)$. For example, we have that $M_{2, 3, 1}$ is non-...
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### open set in tangentspace induces open set in tangentbundle (for homogeneous spaces)

Let $M=G/K$ be a homogeneous space with a $G$-invariant riemannian metric $<.,.>$. Then $G$ defines an action on $TM$ by derivatives. Let $p=eK \in M$. Assuming I have a set $V_p \subset T_pM$ ...
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### Understanding algebraic operations on vector bundles

Let $X$ be a smooth manifold with vector bundles $V$ and $W$. I'm trying to understand how we can construct in general a vector bundle $V \otimes W$. In particular what manifold structure do we give ...
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### Condition for set of de Rham cohomology classes is linearly independent

If we have a set of 1-forms $w_1, ... w_n$ on a smooth manifold $X$ I can show that $w_i$ are linearly dependent if and only if $w_1 \wedge ... \wedge w_n = 0$. I wondered if this is also true in the ...
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### Compact-Open Topology for Space of C^{r} -sections

Given a smooth fibre bundle $\pi: X \rightarrow M$. What is the definition of compact open $C^{r}$-topology on the space of $\mathcal{C}^{r}$-sections?
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### Cohomology of local systems on spaces with the same homotopy type

Suppose we have a homotopy equivalence $f: X \to Y$ with homotopy inverse $g: Y \to X$, and a local system (i.e. a locally constant sheaf) $\mathcal{V}$ on $Y$. In this situation, are any of the ...
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### A lemma in Milnor's book “Topology from the Differentiable Viewpoint”

In preparation for proving the Poincaré-Hopf Theorem, Milnor states in his book (see p. 38, Theorem 1) For any vector field $v$ on $M$ with only nondegenerate zeros, the index sum $\sum \iota$ ...
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### Low torsion in orientable manifolds?

The final sentence on page 170 of Stillwell's Classical Topology an Combinatorial Group Theory is: Poincaré justified the term "torsion" by showing that $(m-1)$-dimensional torsion is present only ...
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### Parallel transport perspective of gauge transformation invariance for connections

Defining a connection on a principal $G$-bundle $P \to M$ is equivalent to defining a parallel transport on $P$ along curves in $M$. With this perspective, Ralph Cohen commented in his notes on the ...
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### What is the tangent space o SO(n) [closed]

It should be the kernel of the map $H\mapsto A^TH+H^TA$ at some $A$ such that $A^TA=I$ . But I cant find this Kernel can someone help me?