Questions tagged [manifolds]
A manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.
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End cohomology and space of ends
I have started learning about end cohomology, and as far as I understand, the zeroth end cohomology $H_e^0(M; \mathbb{Z})$ is isomorphic to the zeroth Čech cohomology $\check{H}^0(e(M); \mathbb{Z})$, ...
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Fundamental group of the homeomorphism group of a compact manifold
Let $X$ be a compact connected manifold and $\mathcal H(X)$ be the group of all homeomorphisms of $X$, equipped with the compact-open topology. Is the fundamental group of $\mathcal H(X)$ countable? ...
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Sheaf cohomology of non-paracompact manifolds (e.g. the long line)
I have long heard that manifolds are "affine". If we allow non-paracompact manifolds, then this seems to fail, since as explained in Dmitri Pavlov's answer, the Serre–Swan theorem fails. I ...
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Does there exist a Dehn filling of an irreducible 3-manifold with toroidal boundaries which is still irreducible?
Let $M$ be a compact, orientable, irreducible 3-manifold with incompressible toroidal boundary (there might be more than one boundary component). Is it always possible to choose appropriate slopes on ...
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LS category of the quotient of a manifold by its involution
$\DeclareMathOperator\cat{cat}$Let $\cat(X)$ denote the Lusternik–Schnirelmann (LS) category of $X$. This homotopy invariant has been well-studied for CW complexes: it is $0$ if and only if $X$ is ...
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Schoenflies problem in PL setting
What is the status of the Schoenflies problem in the PL category? In other words, given an injective PL map $f:S^{n-1} \hookrightarrow S^n$, is it always PL equivalent to the equatorial inclusion? (I ...
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Are Eilenberg-MacLane spaces limits of manifolds?
In this answer, mme shows that for any compact Lie group $G$, there is a model for its classifying space $BG$ which is the direct limit of closed manifolds. If $G$ is discrete (i.e. $\dim G = 0$), ...
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extendability of fibre bundles on manifolds with same dimensions
Let $M$ be an $m$-manifold. Let $M'\subseteq M$, where
$M'$ is also an $m$-manifold.
Let $N$ be an $n$-manifold. Let $N'\subseteq N$, where
$N'$ is also an $n$-manifold.
Suppose there is fibre ...
- 123
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existence of triangulations of manifolds
Let $M$ be a smooth manifold.
Let $K$ be a simplicial complex.
Let ${\rm sd}(K)$ be the sub-division of $K$.
Suppose there exists a simplicial sub-complex $K_1$ of ${\rm sd}(K)$ such that $K_1$ ...
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Is every codimension-one homology class of a closed manifold represented by a $\pi_1$-injective embedded submanifold?
Let $M$ be a connected closed orientable smooth $n$-manifold and $\nu \in H_{n-1}(M, \mathbb{Z})$ a non-trivial codimension-one homology class. It is known that $\nu$ can be represented by an embedded ...
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Why/does 'low-dimension' topology end with dimension 4? [duplicate]
Put another way, assuming it is somewhat fair to say that we (not I, but those who know better--part of my question is whether my stated assumption is in fact warranted) have in some sense a ...
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Proofs of Poincaré duality
I know several proofs of Poincaré duality:
The original proof using dual cell complexes. Probably the nicest version of this uses a handle decomposition.
The argument (in Hatcher and many other ...
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Fundamental group of a generalized connected sum
Let $M$ and $N$ be two $n$ dimensional connected closed manifolds with $n \ge 3$, and let $S$ be a $(n-1)$ dimensional closed submanifold common to $M$ and $N$. Consider the connected sum of $M$ and $...
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Homotopy equivalent but non-homeomorphic high-dimensional manifolds
I have a question motivated by the classification theory of simply-connected closed $4$-manifolds.
Questions: Given any $n\geq 5$, is it possible to find two simply-connected closed $n$-manifolds $M$ ...
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Extending $\mathbb{R}$ to a higher dimensional manifold [closed]
If a topological space $X$ is Hausdorff, connected, second countable, homogeneous (i.e. it has transitive homeomorphism group) and embeds the real line $\mathbb{R}$, does it follow that $X$ is a ...
