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Sep
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comment Examples of n+1D TQFT with 1 dimensional Hilbert spaces on n-torus and n-sphere but higher dimensional Hilbert spaces on other n-manifolds
You may be interested in the work of Dan Freed and Constantin Teleman on invertible field theories. They have theorems that say that a (fully extended) TQFT is invertible whenever the value assigned to certain spheres, or products of spheres are invertible. See e.g. ma.utexas.edu/users/dafr/Aspects.pdf This does not seem to give the result you want directly, but is in the same spirit.
Sep
15
comment Is there an octonionic analog of the K3 surface, with implications for stable homotopy groups of spheres?
In particular, the number 24 appears as the kissing number of the most dense lattice in 4-d. This 24 is surely the same 24, but I haven't thought enough about it to tell a coherent story (maybe such a story involves the binary tetrahedral group). Similarly 240 is the corresponding kissing number for a lattice in 8 dimensions...
Sep
15
comment Is there an octonionic analog of the K3 surface, with implications for stable homotopy groups of spheres?
Probably you are already aware of all this, but this book review written by John Baez contains a lot of interesting numerology that suggests an interesting answer to your question. math.ucr.edu/home/baez/octonions/conway_smith
Sep
11
comment Homotopy types of schemes
The topological space associated to the non-separated ``line with a doubled origin'' is the non-Hausdorff manifold $M=\mathbb C \sqcup_{\mathbb C^\times} \mathbb C$. I think this does not have the weak homotopy type of a finite CW complex. The proof can be adapted from Prop 5.1 in arxiv.org/abs/math/0609665. In this case, we have that $H^2(M) = 0$ by Mayer-Vietoris, but the Hausdorffification is contractible.
Sep
11
comment Homotopy types of schemes
By the way, I expect the etale homotopy type would give a different answer to the analytic homotopy type in general. For example, for $X= \mathbb C^\times$, the analytic homotopy type is that of $S^1 = B\mathbb Z$, but the etale homotopy type (I guess) is $B\widehat{\mathbb Z}$.
Sep
11
comment Homotopy types of schemes
What is Spec of a scheme? I assumed the question was referring to the complex topology, not the Zariski. For a finite type scheme, locally it is cut out by an ideal in $\mathbb C^n$, so I know what to do.
Sep
11
comment Homotopy types of schemes
How do you define the topological space associated to a scheme which is not of finite type?
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Aug
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comment How algebraic is the holonomy map?
@David Roberts: I think the holonomy map is almost never a submersion. The exponential map is not an open map, even for compact Lie groups e.g. $SU(2)$, (see math.stackexchange.com/questions/301504/…), and not surjective in general for noncompact Lie groups (e.g. $SL_2(\mathbb C)$).
Aug
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comment Semisimple monoidal category with duals
An even simpler example: $1 \otimes 1 \simeq 1$, and $1$ is self dual.
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comment Representation Theory of $U(N)$
@André: Fair point! I started to write a more detailed answer, then remembered that I had other things to do... probably this answer should have been a comment.
Jun
20
answered Representation Theory of $U(N)$
Jun
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answered Equivariant Derived Category
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