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Results tagged with gt.geometric-topology
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user 8103
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).
1
vote
Finiteness properties of general topological spaces
Let $X$ be a space, which we may as well assume connected. If $X$ is an ANR, then Milnor gives that $X$ is homotopy equivalent to a countable CW-complex $X'$. The question of when $X'$ is homotopy equ …
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0
votes
Lefschetz duality for twist coefficient
The Lefschetz duality is more naturally expressed as a duality between cohomology and homology. An oriented $n$-manifold with boundary has a fundamental class $[M]\in H_n(M,\partial M;\mathbb{Z})$, an …
- 35.9k
10
votes
Covering manifolds with some other manifolds
A first observation is that such a $k$ may not exist, for example if $N$ does not embed in $M$.
When $N$ is a disk, then $k$ equals the ball category of $M$, denoted $\operatorname{ballcat}(M)$ or $\ …
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8
votes
Triangulations of submanifolds of smooth manifolds
It follows from Verona's solution to Thom's triangulation conjecture that the inclusion $N\hookrightarrow M$ is triangulable whenever it is proper and topologically stable, and $M$ and $N$ are without …
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15
votes
Accepted
What is the 'non-intuitive' part in sphere eversion (turning inside out)?
Watch Outside In (something we should all do anyway, to commemorate Bill Thurston's passing).
To understand the mathematics behind sphere eversions, you should first get a good intuition for the conc …
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11
votes
Which closed orientable $4$-dimensional manifolds cannot be embedded in $6$-space?
$\mathbb{C}P^2$ does not embed in $\mathbb{R}^6$. See
Feder, S.; Segal, D. M.
Immersions and embeddings of projective spaces,
Proc. Amer. Math. Soc. 35 (1972), 590–592.
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2
votes
manifolds with unusual rational cohomology rings
I'm not sure if this is quite what you're looking for, because:
(a) it's a $3$-manifold with boundary, and
(b) it's not the ring structure which is interesting;
but the Borromean Rings link compl …
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2
votes
Accepted
Definition of Milnor exact sequence and complex-oriented generalized cohomology of $\mathbb{...
Milnor's paper is called On axiomatic homology theory and is published in Pacific J. Math.
Volume 12, Number 1 (1962), 337-341. It is also reprinted in Frank Adams' Algebraic Topology: A Student's Gui …
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3
votes
Homotopy Extension Property (HEP)
You need a relative version of the Isotopy Extension Theorem. There is a proof of the non-relative version in Hirsch's "Differential Topology", pp. 177-180. Perhaps more relevant to your question is t …
- 35.9k
1
vote
Identifying the orientation bundle uniquely
Any connected orientable surface $X$ which doubly covers $S$ must be homeomorphic to the orientation cover $\tilde{S}$.
Proof: Connected orientable surfaces are classified up to homeomorphism by th …
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5
votes
Accepted
important vector bundles
The top exterior power of the tangent bundle determines orientablility (the most basic of topological invariants, after dimension) in the sense that $\wedge^nTM$ is a trivial line bundle iff $M^n$ is …
- 35.9k
1
vote
Regular homotopy invariance of Wall's self-intersection form.
I don't know about a rigorous proof, but here is a heuristic explanation you may find helpful.
You have a compact $1$-dimensional manifold given by the self-intersections of the trace $f$ of your reg …
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3
votes
Accepted
Thom's result and Poincaré duality
Let me attempt to answer your question as I understand it.
Let $x\in H^1(M)$ be the Poincaré dual of $[c]\in H_1(M)$ (all (co)homology groups are with $\mathbb{Z}_2$ coefficients). The result of Thom …
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4
votes
Fake projective spaces
The Manifold Atlas page
http://www.map.mpim-bonn.mpg.de/Fake_complex_projective_spaces
has plenty of useful information and references.
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21
votes
Accepted
Link such that deleting any two components leaves an unlink
Yes, this is done in
Penney, D.E., Generalized Brunnian links, Duke Math. J. 36, 31-32 (1969). ZBL0176.22201.
Call a link $(n,k)$-Brunnian if it has $n$ components, and every sublink with $m$ compo …
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