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Results tagged with differential-topology
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user 27004
The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
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
4
votes
1
answer
585
views
Thom space, homotopy group and cohomology group
In Thom's 1952 paper, Thom showed that the Thom class, the Stiefel–Whitney classes, and the Steenrod operations were all related. He used these ideas to prove in the 1954 paper Quelques propriétés glo …
- 10.5k
4
votes
0
answers
239
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Non-spin 5-manifold and $2^2$-Bockstein homomorphism
The $2^2$-Bockstein is $\beta_4$ is associated to
$$0\to\mathbb{Z}/2\to\mathbb{Z}/{8}\to\mathbb{Z}/{4}\to 0,$$
(The $2^n$-Bockstein homomorphism
$$\beta_{2^n}:H^*(-,\mathbb{Z}/{2^n})\to H^{*+1}(-,\m …
- 10.5k
5
votes
1
answer
498
views
Every unorientable 4-manifold has a $Pin^c$, $Pin^{\tilde c+}$ or $Pin^{\tilde c-}$ Structure
The precise statement on J. W. Morgan's "The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds (MN-44)" that 4-manifold $X$ admits a Spinc structure (Lemma 3.1.2) seem …
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7
votes
0
answers
226
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The limitation of $G$ and loop group $\Omega G$ in Atiyah's and Donaldson's work on Instantons
In Atiyah's work [Ref. 1], Atiyah states that "Essentially we shall show (at least for $G$ a classical
group and probably for all $G$) that Yang-Mills instantons in 4D can be naturally identified with …
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5
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answers
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Dimensions of the instanton moduli space from Atiyah-Hitchin-Singer
Atiyah-Hitchin-Singer Ref 1 states that the number of
virtual dimensions of the instanton moduli space
for SU(N) Yang-Mills theory with topological charge $\mathcal{Q}$ over a manifold $X$ is given b …
- 10.5k
5
votes
0
answers
109
views
Induced new structures on Poincare dual manifolds
"R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds (1990)" shows
Given a spin structure on $M^3$, the submanifold $\text{PD}(a)$ can be given a natural induced $\text{Pin}^ …
- 10.5k
5
votes
1
answer
370
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Conversion formula between "generalized" Stiefel-Whitney class of real vector bundles: O(n) ...
$O(n)$ is an extension of $\mathbb{Z}_2$ by $SO(n)$,
$$1\to SO(n) \to O(n)\to \mathbb{Z}_2 \to 1.$$
Below we denote the Stiefel-Whitney class of real vector bundle $V_G$ of the group $G$ as:
$$
w_j( …
- 10.5k
4
votes
1
answer
484
views
Every _______ $d$-manifold has an $S$-structure
I am looking for some analogous nontrivial but known statements and references about statements of the form:
Every _______ $d$-manifold has an $S$-structure.
Here _______ is a placeholder for c …
- 10.5k
6
votes
0
answers
501
views
Yang–Mills existence and mass gap official statement on Euclidean $\mathbb{R}^4$, why not Mi...
Yang–Mills existence and mass gap problem is officially stated by Clay Mathematics Institute:
Yang–Mills Existence and Mass Gap.'' Prove that for any compact simple gauge group G, a non-trivial quant …
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0
votes
0
answers
88
views
Automorphism group of a Lie group $G$ vs that of a double covering group $\tilde{G}$: same o... [duplicate]
Previously there are some counterexamples given in Automorphism group of a Lie group $G$ vs that of a covering group $\tilde G$: same or not? such that the automorphism group of a Lie group 𝐺 is not …
- 10.5k
1
vote
0
answers
137
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Automorphism group of indefinite orthogonal Lie group $G=O(p,q)$ vs that of a double coverin...
Previously I mentioned in Automorphism group of a Lie group $G$ vs that of a double covering group $\tilde{G}$: same or not? that the automorphism group of a Lie group 𝐺 may be the same as that of it …
- 10.5k
0
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259
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Define a characteristic class on a simplicial complex (non-manifold)
Given a simplicial complex with only triangulation and only branching structure, is it enough to define Stiefel–Whitney class?
(Please provide Yes or No answers, and reasonings.)
Given a fixed tria …
- 10.5k
2
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0
answers
109
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Compare two topologies: Three 2-tori inside $S^3 \times S^1 \# S^2 \times S^2$ glued from tw...
We like to ask for the comparison of two topologies of three 2-tori inside the same 4-manifolds glued from two different diffeomorphisms (see the end).
Given an embedded torus $T$ with trivial normal …
- 10.5k
8
votes
1
answer
224
views
Isomorphisms of Pin groups
My goal is to identify the $Pin$ group
$$
1 \to Spin(n) \to Pin^{\pm}(n) \to \mathbb{Z}_2 \to 1
$$
such that $Pin^{\pm}(n)$ are isomorphisms to other more familiar groups.
My trick is that to look at …
- 10.5k