3
votes
0answers
95 views

The monodromy and cut planes.

I am trying to understand the following example from the book Riemann surface by Donaldson (p.48). This was given after introducing the notion of monodromy of the covering. Consider the Riemann ...
3
votes
0answers
80 views

Applications of a theorem of Cartier and Gabriel

In a representation theory course I took we stated and proved the following Theorem due to Cartier and Gabriel: Theorem: Suppose $H$ is a cocommutative Hopf algebra over a field $k$ such that $ ...
3
votes
0answers
159 views

Mackey functor structure on equivariant homotopy groups

I have read that the equivariant stable homotopy groups $\pi_n^{-}(X)=\pi_n(X^{-}) $ of a $G$-space or $G$-spectrum $X$ have a Mackey functor structure. Can somebody please explain how the covariant ...
3
votes
1answer
83 views

Monodromy Representations

Let, be $V$ a connected smooth manifold and $q_1,q_2\in V$ and $F:U\to V$ a connected covering of degree $d$. This covering induces two monodromy representations $\rho_1:\pi_1(V,q_1)\to S_d $ ...
2
votes
0answers
68 views

4D TQFT construction from a modular tensor category

I know the construction of 3D topological quantum field theory (TQFT) from a modular tensor category. I heard that we can even (mathematically) construct 4D TQFT from a modular tensor category. I ...
1
vote
0answers
62 views

2-morphisms from spans of spans

I have a question about the construction of 2-morphisms from spans of spans in the paper "2-vector spaces and groupoid" by Jeffrey Morton . Suppose we have a span of span of groupoids as follows and ...
3
votes
1answer
251 views

Left adjoint and right adjoint/ Nakayama isomorphism

I am reading a paper "2-vector spaces and groupoid" by Jeffrey Morton and I need a help to understand the following. Let $X$ and $Y$ be finite groupoids. Let $[X, \mathbb{Vect}]$ be a functor ...
5
votes
1answer
71 views

Representations of $\pi_1M$ and Heegaard Splittings

I am reading Floer's Instanton-Invariant paper, and am stuck on a sentence. To set the stage: Consider a closed connected oriented 3-manifold $M$ and the nonabelian group $SU_2$. Denote the ...
2
votes
1answer
117 views

Question about Tamafumi Kaneyama's Paper: “On Equivariant Vector Bundles On An Almost Homogeneous Variety”

My reference: http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.nmj/1118795362&page=record I have two question about Proposition 3.3.: Proposition3.3. ...
0
votes
1answer
80 views

Representation of a fundamental group.

Consider the fundamental group $\pi_1(\mathbb{CP}^1\backslash \{a_1, \ldots, a_n\})$. It is said that there is a representation: $\pi_1(\mathbb{CP}^1\backslash \{a_1, \ldots, a_n\}) \to GL(n, ...
5
votes
1answer
439 views

On Frobenius reciprocity theorem

The classical Frobenius reciprocity theorem asserts the following: If $W$ is a representation of $H$, and $U$ a representation of $G$, then $$(\chi_{Ind W},\chi_{U})_{G}=(\chi_{W},\chi_{Res ...
6
votes
3answers
483 views

how to compute the Euler characters of a Grassmannian?

Let $G(n,m)$ be the Grassmannian of all n-dim subspaces of an m-dim vector space over $\mathbb{C}$. How to compute the Euler characters of $G(n,m)$? For example, $G(1, 2)$ is $\mathbb{C}P^1$ which is ...
3
votes
2answers
247 views

what are good references for learning about vector bundles and their sheaves of sections?

I am a beginner in representation theory and algebraic geometry, so that references giving clear explanations of things like the tautological line bundle on $\mathbb P^n$, its dual, and the associated ...