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

Spin bordism with non free involution

Is there a comprehensive account of GEOMETRIC equivariant spin bordism groups with respect to the group Z/2 (instead of homotopy theoretical trough equivariant Thom Spectra), which allow actions which are not free?. I am interested only in low dimensions 0 to 6. There is a computation of the free bordism groups done by Gianvalbo (1976, … Read more

Does a “symbolically elliptic” sequence of operators have an analytic index?

Does a symbolically elliptic sequence of differential operators have an analytic index? cohomology? For example, is there any concrete meaning of the Todd genus of an almost complex manifold in terms of the Dolbeault operator? Consider a smooth manifold $M$ with vectors bundles $E_0$, $E_1$ and a linear differential operator $D\colon C^\infty(E_0)\to C^\infty(E_1)$ on their … Read more

H-principle for smoothing

I’m trying to find algebraic embedding of singular curves into projective varieties that can be smoothed symplectically but not algebraically. It’s not hard (e.g. using the methods in Hartshorne-Hirschowitz “Smoothing algebraic space curve” or Hartshorne’s “Families of Curves in P3 and Zeuthen’s Problem”) to construct such examples for P3 with locally smoothable singularities (even ADE) … Read more

How can one prove that the algebra of smooth functions is semisimple?

I have read in some differential geometry works that the ring of smooth functions C∞(U) is a semi-simple ring, for U⊆Rn an open set; right now I can cite a remark immediately preceding Proposition 2.4.1 on Kostant’s notes “Graded Manifolds, graded Lie theory and prequantisation”. How can one prove such a thing? Is there anywhere … Read more

Visual boundary vs. ideal boundary of hyperbolic manifolds?

I apologize in advance if this is elementary; I have very little experience with hyperbolic manifolds, but I’m using hyperbolic manifolds for part of a current project. Given a discrete torsion-free subgroup of isometries acting on hyperbolic space Γ↪Isom(Hn), then Γ also acts on the boundary sphere Sn−1 of hyperbolic space. As I understand it, … Read more

Equivalence of Flat Fiber Bundles vs Equivalence of Group Actions on the Fiber

Let’s consider all flat fiber bundles with base space $B$ and fiber $F$, where $B$ and $F$ are compact and at least CW-complexes. (perhaps even topological/smooth manifolds if that helps) All those bundles are of the form $$E \cong \widetilde{B} \times_{\pi_{1}} F,$$ where $\widetilde{B}$ is the universal cover of $B$, $\pi_{1}$ is the fundamental group … Read more

Riemannian metrics on a manifold with corners

For a smooth manifold with corners (although maybe there is no universally agreed definition of it), is there always a Riemannian metric making every face totally geodesic? Is there any reference for such a result? Answer AttributionSource : Link , Question Author : UVIR , Answer Author : Community

Reference Request: De Rham isomorphism with Hilbert space coefficients

Let M be a smooth, closed manifold, equipped with a smooth (finite) triangulation K. Further, let H be a Hilbert space, G:=π1(M) and let ρ:G→GL(H) be a representation (with GL(H) denoting the group of (continuous) automorphisms of H). Via the action of ρ, we can therefore regard H as a (left-) C[G]-module with. The triangulation … Read more

Vector bundles on space of germs

Let X be the diffeological space of germs of paths c:R→Rn, where two paths c1,c2 are equivalent if c1(t)=c2(t) for all t in some interval (−ε,ε) (the diffeology on X is the quotient diffeology induced from the diffeological space C∞(R,Rn)). Q: Are all vector bundles on X trivial? Clearly, X is smoothly contractible to a … Read more