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Results tagged with gt.geometric-topology
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user 317
Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
16
votes
1
answer
651
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Elements of a free group that can't be inverted by automorphisms
Let $F_n$ be a free group of rank $n$. Say that $w \in F_n$ is non-reversible if there does not exist any $f \in \text{Aut}(F_n)$ such that $f(w) = w^{-1}$.
Original Question. Intuitively, I expect …
- 63.9k
7
votes
0
answers
258
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Homotopy type of space of embeddings of a disk
Let $M^n$ be a smooth $n$-dimensional manifold and let $\mathbb{D}^n$ be the open unit disk in $\mathbb{R}^n$. Consider the space $\text{Emb}(\mathbb{D}^n,M^n)$ of smooth embeddings of $\mathbb{D}^n$ …
- 44.8k
53
votes
7
answers
10k
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Triangulating surfaces
I've had a few undergraduate students ask me for references for the classical fact (due to Rado) that closed topological surfaces can be triangulated. I know two sources for this, namely Ahlfors's bo …
- 20k
62
votes
9
answers
9k
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Fundamental groups of noncompact surfaces
I got fantastic answers to my previous question (about modern references for the fact that surfaces can be triangulated), so I thought I'd ask a related question. A basic fact about surface topology …
- 2,274
17
votes
1
answer
666
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Relationship between Smith's special homology groups and equivariant homology theory
EDIT: Tyler Lawson's answer was so nice that I was inspired to rewrite the notes discussed below to use Bredon homology in the definition of the Smith special homology groups. The original version is …
- 44.8k
11
votes
2
answers
973
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Subtle point in definition of BNS invariant
Let $G$ be a finitely generated group. Let $S(G)$ be the quotient of $\text{Hom}(G,\mathbb{R}) \setminus \{0\}$ by the equivalence relation that identifies two homomorphisms if they differ by scaling …
- 1,713
20
votes
2
answers
3k
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First Chern class of a flat line bundle
A referee asked me to include a reference or proof for the following classical fact. It's not hard to prove, but I'd prefer to just give a reference -- does anyone know one?
Let $X$ be a nice space …
- 44.8k
12
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0
answers
784
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Commutator subgroup of a surface group
Let $\Sigma_{g,n}$ denote a compact orientable genus $g$ surface with $n$ boundary components. Assume that $g \geq 1$ and fix a basepoint $p \in \Sigma_{g,n}$. Define $S \subset [\pi_1(\Sigma_{g,n}, …
- 44.8k
7
votes
2
answers
441
views
Fragmenting a homeomorphism of a compact manifold
Let $M$ be a compact manifold and let $f : M \rightarrow M$ be a homeomorphism which is isotopic to the identity. We will say that $f$ can be fragmented if it satisfies the following property. Let $ …
- 12.8k
14
votes
3
answers
979
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Homotopy type of set of self homotopy-equivalences of a surface
Let $\Sigma$ be an oriented topological surface. For simplicity, assume that the genus of $\Sigma$ is at least $2$. There are a number of classical results on the homotopy types of various groups of …
- 20k