Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with gt.geometric-topology
Search options not deleted
user 8032
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).
0
votes
First usage of the terms pseudo-isotopy and concordance in manifold theory
MathSciNet refers to a paper of Bing from 1959 with the term "pseudo-isotopy" in the math review:
Bing, R. H.
The cartesian product of a certain nonmanifold and a line is
E^4. Ann. of Math. (2)70(1959 …
- 18.5k
5
votes
Accepted
Is the Artin Spin construction related to the suspension functor?
This question is answered in section 4 of my first paper (with Alex Suciu)
Klein, John R.; Suciu, Alexander I.
Inequivalent fibred knots whose homotopy Seifert pairings are isometric.
Math. Ann. 289 ( …
- 4,588
9
votes
Accepted
Atiyah duality without reference to an embedding
Here is another short construction which is much simpler and just takes a few lines.
Let $M$ be a closed $n$-manifold. Consider the diagonal $M \to M \times M$. It is an embedding. Take its Pontryagi …
- 18.5k
7
votes
Atiyah duality without reference to an embedding
Assume $M$ is a closed, smooth manifold of dimension $n$. Let $\tau^+$ be the fiberwise one point compactification of its tangent bundle. This is a fiberwise $n$-spherical fibration equipped with a p …
- 18.5k
9
votes
Compelling evidence that two basepoints are better than one
I just wanted to add something to the discussion about the utility of adding additional basepoints. It turns out this is crucial for understanding
certain aspects of embedding theory. See the bottom o …
- 244
8
votes
What characteristic class information comes from the 2-torsion of $H^*(BSO(n);Z)$?
See
Brown, Edward, The cohomology of $B\text{SO}_n$ and $B\text{O}_n$ with integer coefficients. Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–288
for an answer to your questions. (As for applications: …
- 18.5k
14
votes
How to prove the connected sum of two closed aspherical n-manfolds (n >2) is not asperical?
Here's a different way to see it. Let $M$ and $N$ be aspherical of dimension at least 3.
Then the wedge $M \vee N$ is aspherical (but not a manifold). Let $M\sharp N$ be the connected sum. Then we get …
- 18.5k
14
votes
Unstable manifolds of a Morse function give a CW complex
(1). Some experts tell me that Laudenbach's paper is incomplete and contains gaps.
I will retract this for now. I do recall being told this, but I am not
aware at this point in time where the gaps …
- 18.5k
17
votes
Accepted
Immersions of surfaces in $\mathbb{R}^3$
(Based on the comment of Mariano Suárez-Álvarez, there was a false assumption in my original answer. This is an attempt to correct it.)
1) Let $M$ be a closed smooth manifold with $k < n$. According …
- 18.5k
3
votes
Accepted
Wall self-intersection invariant for odd-dimensional manifolds?
I doubt that what you are proposing as the receptacle for the obstruction is the correct abelian group. For one thing, you are not taking into account the involution on the canonical double cover of t …
- 18.5k
6
votes
Can one disjoin any submanifold in $\mathbb R^n$ from itself by a $C^{\infty}$-small isotopy?
There are lots of counter-examples.
Here's one:
The fibration $\text{SO}(3) \to \text{SO}(4) \to S^3$ splits, since it is the principal bundle of the tangent bundle of $S^3$, and the latter is paral …
- 18.5k
10
votes
Accepted
The space of contractible loops of a finite dimensional $K(\pi,1)$
The statement is true for a $K(\pi,1)$ but not true for other $X$. Finite dimensionality is not relevant.
Here's a sketch: let $X = K(\pi,1)$ and I will assume $X$ has the homotopy type of a CW com …
- 18.5k
11
votes
Accepted
Linking topological spheres
Let $C = S^3 \setminus A$. Alexander duality says that
$$
H_1(C) \cong H^1(A) \cong \Bbb Z\, .
$$
Let $\alpha: S^1 \to C$ be any map representing a generator of $H_1(C)$ (every first homology class …
- 18.5k
12
votes
1
answer
209
views
A variant of $\ell^2$-cochains
Suppose $X$ is an infinite countable CW complex which satisfies the following property: for all $k$-cells $e$, the number of $(k+1)$-cells incident to $e$ is at most $c_k$, where the latter is some n …
- 924
2
votes
Accepted
Embedding spaces and surface knots in high dimensional manifolds
Here are some general comments. We can let $\Sigma$ be any closed smooth $k$-manifold and let $X$ be any smooth $n$-manifold.
Fix a basepoint embedding $\Sigma \to X$.
Let $N$ be a compact regular n …
- 18.5k
7
votes
Manifolds covered by a single disc
The answer (to the first question) is yes in the smooth case: If $M^m$ is closed and compact, then there is a Morse-Smale function on $M$ with a single critical point of index $m$. Work by Lizhen Qin …
- 643
10
votes
classifying space of orthogonal groups
$BO$ can be defined as the colimit over $(k,n)$ of Grassmanians $G_k(\Bbb R^n)$ of $k$-dimensional linear subspaces of $\Bbb R^n$ (the limit over $n$ is defined by standard inclusions $\Bbb R^n \subse …
- 18.5k
2
votes
Ehresmann fibration theorem for manifolds with boundary
Let $D(M)$ be the boundary of $M \times [0,1]$ (by smoothing corners, this can be understood as smooth). Then $f: M \to N$ induces a smooth map
$$
D(f): D(M) \to D(N)\, .
$$
Further, $D(f)$ is a prop …
- 18.5k
8
votes
contractible configuration spaces
I believe each of these arguments will work.
Argument 1: Consider $S^n \subset \Bbb R^{n+1} \subset S^{n+1}$, where the last inclusion is
given by the upper hemisphere (which is homeomorphic to $\Bbb …
- 18.5k
7
votes
Homotopy groups of spaces of embeddings
Here are some comments:
1) Concerning finiteness results for spaces of embeddings, here is what I remember. The layers of the Goodwillie-Weiss tower when $M^m$ is closed and $N= \Bbb R^n$ have finit …
- 18.5k
5
votes
When does an even-dimensional manifold fiber over an odd-dimensional manifold?
No, I don't believe there's a simple solution. But
here's an approach to the problem which indicates how it can be fractured up.
Assume $M,N$ are closed and connected.
If $f\: M \to N$ is homotopic …
- 18.5k
14
votes
Accepted
Characteristic classes for block bundles
I don't know where the results are written down in one place (perhaps in the book of Madsen and Milgram?), but see the the end of this post for a list of references.
In any case, here is a proof of …
- 18.5k
11
votes
Intuition behind Alexander duality
Yet another way to get it. Alexander duality is closely related to Poincare duality: suppose we can write
$$
S^n = K \cup_A C
$$
where $K$ and $C$ are codimension zero compact manifolds with common bo …
- 18.5k
6
votes
Searching for an unabridged proof of "The Basic Theorem of Morse Theory"
My recollection is that Milnor's proof gives exactly what you are asking. In fact, see the remark on the bottom of page 17 of his book.
- 18.5k
2
votes
Has this kind of question in topology a special name?
How about $\pi_0(\text{Homeo}(X))$?
The papers of Weiss and Williams (automorphisms of manifolds and algebraic $K$-theory...) are relevant since they reduce computations of $\pi_i(\text{aut}(X))$ for …
- 18.5k
11
votes
Accepted
How does the Framed Function Theorem simplify Cerf Theory?
The framed function theorem tells you that up to "contractible choice" a compact manifold admits a framed function: i.e., a function as you prescribe. Furthermore, a framed function is supposed to giv …
- 18.5k
1
vote
Regular homotopy invariance of Wall's self-intersection form.
Even more is true in the context of surgery theory: let
$q \ge 3$ with $q$ odd, assume $M$ is a $1$-connected closed smooth manifold of dimension $2q$. Let $$I^{\text{fr}}_q(M)$$ denote the space of i …
- 18.5k
2
votes
What should be taught in a 1st course on smooth manifolds?
I do have one addition to make to the above. At our university we usually use a combination of Guillemin and Pollack and Milnor. There is another approach at a first course which some have found usefu …
- 18.5k
18
votes
What is the intuition behind the Freudenthal suspension theorem?
The Freudenthal theorem is really a special case of the phenomenon called "homotopy excision" aka the Blakers-Massey triad theorem. The idea is that one has an inclusion
$$
(C_-X,X) \to (\Sigma X,C_+X …
CommunityBot
- 1
0
votes
Poset fiber theorems under a special assumption on the poset map?!
In the following, I shall assume that the posets $P$ and $Q$ are finite.
Then it is at least true from the condition that $f^{-1}(q)$ is contractible for all $q \in Q$ that the map $|f|$ is a homolog …
- 18.5k
5
votes
Homotopy type of the plane minus a sequence with no limit points
Here is a more naive solution, as least if the sequence is countable. Let
$\Bbb N \subset \Bbb R^2$ be the embedding defined by the sequence.
Then there is an isotopy from this embedding to the stand …
- 18.5k
6
votes
Stratification of smooth maps from R^n to R?
It looks to me that what you are really interested in is the Thom-Boardman stratification of the function space. For that I would recommend the well-written, Stable Mappings and Their
Singularities b …
- 18.5k
5
votes
Accepted
When does an antipodal map on a manifold extend to the antipodal map on a spheres
Let me elaborate on my comment above. Suppose $M$ is a manifold equipped with a smooth $\Bbb Z_2$ action that is also free. Then there is an equivariant smooth embedding $M \to S^j$, for some $j$, whe …
- 18.5k
10
votes
Can we decompose Diff(MxN)?
When $N$ is the circle there's a sort of answer. In fact there's a whole chapter in the book
Burghelea, Dan; Lashof, Richard; Rothenberg, Melvin: Groups of automorphisms of manifolds. With an append …
- 18.5k
11
votes
topological type of smooth manifolds with prescribed homotopy type and pontryagin class
In the $1$-connected case, one may argue as follows:
Let $X$ be a closed $1$-connected smooth $n$-manifold, $n \ge 5$. The theory of the Spivak fibration shows that any homotopy equivalence $f: M^n …
- 18.5k
12
votes
Accepted
"C choose k" where C is topological space
I presume you are asking is whether one can make sense of $\binom{X}{k}$ as a space in such a way that it relates to the formula you are asking about.
The answer is yes.
First let $l = 1$. For a sp …
- 18.5k
12
votes
The space of framed functions
Eliashberg once told me that the framed function theorem should be a consequence of his work on wrinkled maps.
Igusa and I gave a fairly direct proof that the space of framed functions on the circle …
- 18.5k