Higher tangent bundles of manifolds with non integer dimension

One way to define the tangent space of a manifold at a point p∈M is the following: We define an equivalent relation on the space of curves passing p as follows: Two curves α,β are equivalents iff they have tangencity of order at least one that is ∥α(t)−β(t)∥=o(|t|), in a local smooth coordinate. Then the … Read more

Relationship between Gaussian and bisectional curvature

Let f:CP1→X be an smooth embedding which its image is the curve C where X is a Kähler manifold. Do we have that supCP1K(f∗ω)<supCK(ω),where K(f∗ω) denotes the Gaussian curvature of pullback of metric and K(ω) is the bisectional curvature? Answer AttributionSource : Link , Question Author : user44803 , Answer Author : Community

Homotopy invariance of the moduli stack of smooth $G$-bundles?

This question ought to have a straightforward (perhaps even glaringly obvious) answer, but so far I’ve already wasted a few hours trying to untangle this web of inconsistent identifications. I’m sure someone more experienced in this area will be able to quickly point out exactly where things go wrong. Let $G$ be a Lie group … Read more

English language and Mathematics

I have a question maybe more relevant to an English language section of StackExchange, but I doubt that anybody but a Mathematician could properly answer my question. Let $\mathcal M$ be a smooth manifold, let $X$ be a smooth vector field on $\mathcal M$ and let $\Sigma$ be a smooth hypersurface of $\mathcal M$. Let … Read more

Seiberg-Witten theory in 4d is categorification of Seiberg-Witten in 3d

According to Gukov et al. in this 2017 paper Seiberg-Witten theory in 4d categorifies Seiberg-Witten theory in 3d. In what sense is this phrase mentioned? I know what the process of categorification is (e.g. how Khovanov homology categorifies Jones polynomial). What is the exact relation between the 3d and 4d versions of Seiberg-Witten theory and … Read more

A quantity associated with a Riemannian surface

Assume that $E$ is a Riemannian vector bundle, then its structure group is reduced to $O(n)$. Then the structure group of $E \oplus E$ is reduced to $D(O(n) \oplus O(n)) \subset Sp(2n)$ where $$D(O(n) \oplus O(n))=\{ A\oplus A\mid A \in O(n)\}$$ So there is a natural symplectic structure $\omega $ on each fiber of $E … Read more

What is the good notion of supervariety?

The principal (I think) difference between the notions of manifold in differential (including complex analytic) topology and in algebraic (or especially arithmetic) geometry is that for the former the “local models” are the same while for the latter they may vary. This somehow suggests that there may be a version of the notion of supervariety … Read more

A cohomology associated to a symplectic manifold

Let (M,ω) be a symplectic manifold. Let Ωkω(M)={α∈Ωk(M)∣α∧ωis an exact form} Then we have a chain comlex…→Ωk−1ω(M)→Ωkω(M)→Ωk+1ω(M)… with the standard differential operator d. so we obtain a cohomology Hkω. Does this cohomology depend on choosing the symplectic structure ω? Does this cohomology contains some information about the symplectic manifold? Is there a trivial description for … Read more

Lower bound on ϵ\epsilon-covers of arbitrary manifolds

Let M⊂Rd be a k-dimensional manifold embedded in Rd. Let N(ϵ) denote the size of the minimum ϵ-cover P of M, that is for every point x∈M there exists p∈P such that ‖. What is known about lower bounds on \mathcal{N}(\epsilon)? If M were d-dimensional then there is a simple volume argument that shows that … Read more

Relating bordism invairants in dd and d+2d+2 dimensions

Are there some relationship between mapping the bordism invairants of eq.1 and eq.2? ΩdO(B(PSU(2n)⋊ and \Omega_{O}^{d+2}(K(\mathbb{Z}/{2^n},2)) \tag{eq.2} Here K(G,2) is the Eilenberg–MacLane space. We can take d=3 here. The semi-direct product of q \in \mathbb{Z}_2 here acts on the g \in PSU(2^n) as the complex conjugation * and transpose T, so q g q= g^{*T} … Read more