# Tagged Questions

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### Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
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Since the complex Grassmannian $G_k(\mathbb{C}^n)\cong SU(n)/S(U(k)\times U(n-k))$ is connected and simply connected, the first two de Rahm cohomology groups are given by $$... 0answers 35 views ### Homology of smooth loop spaces of spheres I'm looking for a reference on the homology of C^\infty(S^1,S^n). So far I've only been able to find references for spaces of maps which are C^k or in a Sobolev class. I also have references which ... 2answers 69 views ### What is the calculus-theoretic formula to calculate the homotopy class/degree of a map T^2\to S^2? I know by Hopf classification theorem that [T^2;S^2](torus to sphere) are classified by the integral cohomology group H^2(T^2;\mathbb{Z})\approx\mathbb{Z}. Also I understand that in general, given ... 0answers 42 views ### Using the Hodge theorem to decompose the metric tensor There has been a previous discussion about concrete constructions using the Hodge theorem , Construction of Hodge decomposition Let me try to ask here about a specific case where one is trying to ... 1answer 77 views ### Monodromy representation of Airy equation Let K=\Bbb{C}(z) with the usual derivation and consider the Airy dierential equation y^{(2)}-zy=0. How to determine the monodromy representration? Airy equation is not Fuchsian diferential ... 0answers 68 views ### differential forms and orientation in Bott and Tu I am reading Bott and Tu book on my own. If you have the text you can answer my question perhaps, if not it would be too long to explain it. I have gotten to page 29. I am confused about proposition ... 0answers 36 views ### Minimum regularity Of Stoke's theorem to hold in smooth manifold. Stokes’ Theorem on Manifolds is often express as follows: Given a differential m-form \omega whose support is the C^{\infty} m-dimensional compact manifold {\cal{M}} with boundary ... 1answer 64 views ### Decomposition of cohomology group on S^{n} If we have decomposition of cohomology group on S^{n} it looks like H^{n}(S^{n})=H^{n}(S^{n})_{+}\oplus H^{n}(S^{n})_{-}, where H^{n}(S^{n})_{\pm} cohomology of invariant or anti-invariant n ... 1answer 34 views ### Winding number from complex analysis and differential geometry I showed that for a differentiable function f:S^1 \rightarrow S^1, the winding number is given by \frac{1}{2 \pi i } \int_{S^1} \frac{f'(z)}{f(z)} dz. Now I want to show that the winding number ... 1answer 56 views ### the relation between cohomology and homomorphism I meet a problem, how can I understand H^1(M,\mathbb{R})\cong Hom(\pi_1(M),\mathbb{R})? Where M is a compact manifold. Thanks in advance. 0answers 36 views ### Sections of the dual bundle of a smooth vector bundle Let M be a smooth manifold and E,F,G smooth vector bundles over M. Denote the global sections by \Gamma(.). In this question, it is proven that the canonical map ... 1answer 62 views ### De Rham cohomology for \mathbb{R^2} De Rham cohomology groups for \mathbb{R^2}. H^{0}_{dR}(\mathbb{R}^{2})=\mathbb{R} since Z^{0}(\mathbb{R}^{2}) is the one dimensional space of locally constant functions on \mathbb{R}^{2} and ... 1answer 128 views ### Relations of Characteristic classes: Chern, Stiefel-Whitney, Pontryagin, Euler, Wu class. [closed] We have a quite some characteristic classes: Chern class, Stiefel-Whitney class, Pontryagin class, Euler class, Wu class, etc. I wonder whether some math experts can use simple words and basic ... 2answers 222 views ### Spin manifold and the second Stiefel-Whitney class We know that: Spin structures will exist if and only if the second Stiefel-Whitney class w_2(M)\in H^2(M,\mathbb Z/2) of M vanishes. Can someone use simple words and logic to show why the ... 0answers 127 views ### Which manifolds are zero sets of \mathbb R^n valued maps If M is a smooth manifold, then any framed submanifold N is the preimage f^{-1}(y)  for a smooth sphere-valued map f transversal to y, with the framing of the normal bundle induced by f. ... 1answer 133 views ### Another differential topology lemma Another lemma (1) Why can we assume z=f(z)=0 and that U is convex? (the coordinate domains of the manifolds can be taken to be balls?) (2) Why is it enough to consider the special case of a ... 1answer 166 views ### A differential topology lemma Consider the following lemma (1) How come he talks about degrees here, after all he doesn't assume X to be oriented? (2) Why is \bar{v}|\partial X homotopic to g? (NOTE: we consider them as ... 2answers 83 views ### Homology of manifolds with boundary If M is a compact topological manifold WITH boundary does it follow that its homology groups are finitely generated and zero almost all of them? I know it is true in case it has no boundary (i.e. is ... 0answers 36 views ### The Morse complex of a manifold with boundary For a smooth manifold with boundary M and \partial M = V_+ \cup V_- two disjoint sets of boundary components, one usually defines the Morse complex of M using a Morse-Smale pair (f,X) such ... 1answer 62 views ### Tangent bundle of P^n and Euler exact sequence I'm thinking about the Euler exact sequence for complex projective space, and I'm a bit confused. In the topological category, one has$$ T\mathbb{P}^n \oplus \mathbb{C} \cong L^{n+1} $$where L ... 1answer 27 views ### Alexander duality formulation + Jordan-Brouwer separation In Davis & Kirk LNAT p.71 there is written: (1) How does this imply the Alexander duality \tilde{H}^k(A)\cong \tilde{H}_{n-k-1}(\mathbb{S}^n\!\setminus\!A)? (2) Is it assumed that the ... 2answers 100 views ### Chern-Weil: why do we divide by 2\pi? So here's a somewhat incoherent question. To define characteristic classes in the Chern–Weil way, one takes a curvature form \Omega on a vector bundle E \to M and an invariant polynomial ... 0answers 17 views ### spin^c structures and charged spinors Given a spin structure and a complex line \mathcal{L} we can form the tensor product of the complex spinor bundle S and this line S\otimes\mathcal{L}. A spin^c structure attempts to construct ... 0answers 25 views ### express skew-commutative product of Whitney sum of vector bundles in tensor products Let \xi and \eta be vector bundles over a paracompact space B and \xi\oplus\eta be their Whitney sum. Can we write \Lambda(\xi\oplus\eta)\cong \Lambda(\xi)\otimes \Lambda(\eta) as (graded) ... 1answer 203 views ### Lie groups as manifolds In Weinberg's Classical Solutions in Quantum Field Theory, he states Lie groups, such as SU(2) or SO(3), may be viewed as a manifold. My questions are, If we can interpret, e.g. SU(2) as a ... 3answers 127 views ### let \xi be an arbitrary vector bundle. Is \xi\otimes\xi always orientable? Let \xi=(E,p,B) be a line bundle (not nec. orientable). Then the tensor product \xi\otimes \xi is orientable. I obtain this by choosing b\in U\cap V, U,V open in B such that ... 2answers 106 views ### tensor product of two vector bundles Let \xi and \eta be two vector bundles over the same base space B. Then \xi\otimes \eta is orientable if and only if \xi and \eta are both orientable. How to prove this true or not true? ... 0answers 48 views ### Can Vanishing Cycles be Described as Fibers over Critical Points in a Lefschetz Fibrations? I'm trying to see if it makes sense to see vanishing cycles in a Lefschetz fibration as the fibers over critical points. A Lefschetz fibration f: M^4 \rightarrow X , where M^4 is a smooth ... 0answers 13 views ### Orthogonal Array Planes In an OA-plane (Orthogonal Array-Plane) how we define the touching of two lines? If we consider OA of strength of 3 with unit index , that is the plane OA-plane of rank 3 (we call this Laguerre ... 1answer 30 views ### Is there a name for the result that homotopy groups are insensitive to removing submanifolds of sufficiently large codimension? First some motivation. Consider \mathbb{R}^n-\{0\}. This is simply connected iff n > 2, since it deformation retracts to S^{n-1}. If instead we consider \mathbb{R}^n - L where L is a ... 3answers 160 views ### Is it possible a trivial fiber bundle with nonzero holonomy? Let P\rightarrow M be a principal bundle with structure group G. Suppose that the bundle is trivial M\times G; is it possible to have a nonzero holonomy along some closed trajectory on M for ... 1answer 133 views ### Uniqueness of curve of minimal length in a closed X\subset \mathbb R^2 Suppose X is a simply connected closed subset of \mathbb R^2. Let a,b belong to X. Is it true that there is at most one curve inside X from a to b such that the length of the curve is ... 1answer 117 views ### How can I understand the three-dimensional space forms? Here is what I know: A space form is defined as a manifold admitting a Riemannian manifold of constant sectional curvature A classical result of Cartan states that a manifold is a space form if and ... 1answer 82 views ### De Rham cohomology of T^*\mathbb{CP}^n I am a bit rusty on my de Rham cohomology, and I'm hoping that someone here could help me. I want to find the cohomology of T^*\mathbb{CP}^n (seen as a real manifold). Now, this should be equal to ... 1answer 60 views ### Relation between different definitions of degree in complex geometry Consider a holomoprhic map from a Riemann surface$$ f: \Sigma_g \to \mathbb{CP}^n.  This is given by some homogeneous polynomials in some variables. How can we show that the homogeneous degree $d$ ...
Let $B$ be a compact Hausdorff space and $C^0(B)$ be the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$, let $\Gamma(\xi)$ denote the $C^0(B)$-module consisting ...