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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).
13
votes
Where is the Steenrod Realization problem at?
A lot depends on what you want to know about realizability. You could argue that Thom's paper settles the problem: in the mod 2 case, every homology class is realizable by a map; in the integral case, …
- 12.9k
7
votes
Realizing integral homology classes on non-orientable manifolds by embedded orientable subma...
Here are some comments that don't really answer the question, but are too long for the comment box.
Firstly, the Poincaré dual of $\nu\in H_n(M;\mathbb{Z})$ is a twisted integer class $D\nu\in H^{m-n} …
- 35.9k
14
votes
Integral homology classes that can be represented by immersed submanifolds but not embedded ...
This is a great question, and I don't have an answer but this is too long for a comment.
Working mod $2$, a codimension $k$ homology class $z\in H_{m-k}(M;\mathbb{Z}/2)$ is realizable by an embedding …
- 35.9k
9
votes
Accepted
Integral homology classes of which no multiples admit embedded representatives with trivial ...
With the trivial normal bundle condition, it's fairly easy to produce non-realizable examples using Théorème II.2 of Thom's paper. Namely, a class $z\in H_l(M^n;\mathbb{Z})$ is realizable by an embedd …
- 35.9k
7
votes
Equivariant cohomology of the complement to the arrangement $\bigcup_{i\neq j}\vec x_i = \ve...
$\DeclareMathOperator\Conf{Conf}\DeclareMathOperator\SO{SO}$Here is a partial answer, which at least illustrates how to attack these problems using the methods of algebraic topology.
As usual, to comp …
- 1,446
7
votes
Accepted
on second cohomology of $S^1$-manifold
Yes. This follows from the Leray-Serre spectral sequence of the fibre bundle
$$
M\to M_{S^1} \to BS^1
$$
which has $E_2^{p,q}=H^p(BS^1;H^q(M;\mathbb{Z}))$ and converges to (the associated graded of th …
- 35.9k
16
votes
Can the nth projective space be covered by n charts?
It seems worth giving the cup-length argument, as it's relatively short and sweet.
Suppose $\mathbb{R}P^n=U_1\cup\cdots\cup U_n$, with each $U_i\approx\mathbb{R}^n$, and let $c\in H^1(\mathbb{R}P^n;\m …
- 35.9k
6
votes
Accepted
Künneth formula and induced map in homologies
Here is an example which will not make you very happy. There is a degree one map $f:S^2\times S^1 \to S^3$ which just collapses the complement of an embedded open disk. Take $a\in H_2(S^2;\mathbb{Z})$ …
- 35.9k
22
votes
Is there a $4$-manifold which Immerses in $\mathbb{R}^6$ but doesn't Embed in $\mathbb{R}^7$?
Edit: The answer below is incorrect. In fact, $\bar{w}_3(\mathbb{R}P^2\times\mathbb{R}P^2)=0$ (thanks to Rafal Walczak for pointing this out) so by the cited result $\mathbb{R}P^2\times\mathbb{R}P^2$ …
- 35.9k
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 …
- 35.9k
5
votes
Accepted
Reference for Cochran-Orr-Teichner's filtrations on knot concordance
There are summarys of parts of Cochran, Teichner and Orr's paper in:
These lecture notes of Peter Teichner, typed up by Julia Collins and Mark Powell;
Mark Powell's 2011 Edinburgh PhD thesis;
Julia C …
- 35.9k
8
votes
Accepted
Homotopy in $X$ and homology in $X \times I$
You are talking about the notion of L-equivalence, studied by Thom in his seminal paper
Thom, René, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28, 17-86 (1954). …
- 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 $\ …
- 35.9k
18
votes
Accepted
Wu formula for manifolds with boundary
A relative Wu formula for manifolds with boundary is discussed in Section 7 of
Kervaire, Michel A., Relative characteristic classes, Am. J. Math. 79, 517-558 (1957). ZBL0173.51201.
In particular, t …
- 35.9k
1
vote
Embeddings without nonvanishing normal vector fields
I gave an orientable example in my answer to this later question. In particular, any embedding $\mathbb{C}P^2\hookrightarrow \mathbb{R}^7$ will have this property.
- 35.9k