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Results tagged with gt.geometric-topology
Search options questions only
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user 27004
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).
21
votes
2
answers
912
views
Define the 3d Chern-Simons TQFT on a discrete simplicial complex
Question: What is the challenge and the current status to define the 3d Chern-Simons(-Witten) (CSW) theory on a simplicial complex or on a discrete lattice? (Or is there a no-go or an obstruction behi …
- 10.5k
13
votes
1
answer
882
views
Critical dimensions D for "smooth manifolds iff triangulable manifolds"
I am aware that at least for lower dimensions,
"smooth manifolds iff triangulable manifolds"
at least for dimensions below a certain critical dimensions D.
My question is that for
For orientable m …
- 10.5k
10
votes
1
answer
601
views
Non-triangulable 4-manifold as a boundary of some 5 manifold
We know that there are non-triangulable 4-manifolds, such as the E$_8$ manifold.
Can E$_8$ manifold be a boundary of some 5-manifold $M_5$? Can such a $M_5$ be triangulable or non-triangulable? What …
- 10.5k
10
votes
1
answer
372
views
Discrete Pin structures
It is clear that an oriented manifold $M^n$ (with dimension $n$) admits spin structures if and only if its second Stiefel-Whitney class $[w^2]\in H^2(M,\mathbb Z_2)$ vanishes. In the construction of t …
- 10.5k
9
votes
2
answers
628
views
Künneth formulas/theorem for bordism groups and cobordisms?
We are familiar with Künneth theorem:
The Kunneth formula is given by $R$ as a ring, $M,M'$ as the R-modules, $X,X'$ are some chain complex. The Kunneth formula shows the cohomology of a chain comple …
- 10.5k
9
votes
1
answer
524
views
Does there exist a discrete gauge theory as a TQFT detecting the figure-8 knot?
My question: Does there exist a discrete gauge theory as TQFT detecting the figure-8 knot?
By detecting, I mean that computing the path integral (partition function with insertions of the knot …
- 10.5k
9
votes
1
answer
566
views
Oriented bordism in higher dimensions (e.g. $12 \leq d \leq 28$)
The classification of oriented compact smooth manifolds up to oriented cobordism is one
of the landmarks of 20th century topology. The techniques used there form the part of the foundations of differe …
- 10.5k
8
votes
0
answers
196
views
"Gerbes" in the cobordism theory
In a lecture I attended today, I heard the use of gerbes in the cobordism theory.
Previously, I use cobordism theory, but I never encounter the term "gerbes" when I work on bordism or cobordism group …
- 10.5k
8
votes
1
answer
565
views
Majorana modes and the first Stiefel–Whitney class
The first Stiefel–Whitney class of a vector bundle is an element in the first cohomology group of the base space. Namely, the first Stiefel–Whitney class for a vector bundle $E$ over a $d$-dimensiona …
- 10.5k
8
votes
1
answer
1k
views
Cobordism Theory of Topological Manifolds
Unfortunately, due to my ignorance, my present knowledge is limited to the cobordism Theory of Differentiable Manifolds.
Cobordism Theory for DIFF/Differentiable/smooth manifolds
However, there ar …
- 10.5k
8
votes
2
answers
2k
views
Any 3-manifold can be realized as the boundary of a 4-manifold
We know
"Any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$." See this post: Elegant proof that any closed, oriented 3-manifold is the boundary of some orient …
- 10.5k
7
votes
0
answers
226
views
The limitation of $G$ and loop group $\Omega G$ in Atiyah's and Donaldson's work on Instantons
In Atiyah's work [Ref. 1], Atiyah states that "Essentially we shall show (at least for $G$ a classical
group and probably for all $G$) that Yang-Mills instantons in 4D can be naturally identified with …
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7
votes
0
answers
1k
views
Applications of E8 manifold
The $E_8$ Cartan matrix is given by,
$$
K_{E_8}=\begin{pmatrix}
2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\
-1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\
0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & 2 …
- 10.5k
7
votes
1
answer
258
views
Bordism for oriented triangulable manifolds without smooth differentiable structures
We know the bordism group for oriented smooth differentiable structures such as $\Omega_d^{SO}$ that requires the special orthogonal group structure on the tangent bundle $TM$ of manifold $M$.
$$\Omeg …
- 10.5k
7
votes
2
answers
690
views
Variants and Generalizations of Arf (-Brown-Kervaire) invariants
(1) I encounter the Arf invariants in Kirby-Taylor, Pin structures on low-dimensional manifolds. The form that I looked at was:
$$
S(q)=|H^1(M^2,\mathbb{Z}_2)|^{-1/2} \sum_{x\in H^1(M^2,\mathbb{Z}_2)} …
- 10.5k