1
vote
1answer
45 views

Symplectic submanifolds in $\mathbb{R}^{4}$

Which symplectic submanifolds can be realized in $\mathbb{R}^{4}$? It easy to show that such submanifolds aren't compact. So, they are spheres with some handles and holes. Which relations between the ...
0
votes
0answers
23 views

History of vectorial bundles in articles or papers?

I'm looking for an article or book that gives a thorough and interesting history of bundles and vectorial bundles in algebraic topology. I'm looking for it for my own learning, please help its ...
1
vote
1answer
39 views

Tangent bundle of a manifold [duplicate]

can anyone help me with this problem: Show that for a manifold $M$, the tangent bundle $TM$ also has the structure of a manifold. If $M$ is an n-manifold, what is the dimension of $TM$? for the 1st ...
1
vote
1answer
35 views

Construction of Hodge decomposition

We know Hodge decomposition splits any $k$-form into three $L^2$ components. And I see some proofs, none of them provide an explicit constructive method. Is there any general method to construct one? ...
0
votes
0answers
40 views

English edition of Vol 9 of Dieudonné's Foundations of Modern Analysis?

I have found the first 8 volumes of Dieudonné's Foundations of Modern Analysis in English translation, but I'm having difficulty locating volume 9. I have searched the catalogues of numerous libraries ...
2
votes
1answer
51 views

vector bundles and their cross-sections

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 ...
2
votes
0answers
29 views

Reference about history of characteristic classes

I'm looking for a good reference about history of characteristic classes. For example, I would like to know how Chern define the Chern class at first in history, or who rewrote the characteristic ...
0
votes
0answers
37 views

References Request (Poincare Duality via Unimodular pairing, de Rham isomorphism)

Chapter 0 of Griffith's Harris contains a substantial amount of topological results that I have not seen elsewhere, namely: Poincare duality as a unimodular intersection pairing on homology. Also ...
0
votes
1answer
69 views

compact open topology on M(X,Y) continuous function space

I'm really confused with the statement. let $X$ be a set endowed with the discrete topology and let $Y$ be any topological space. Show that $M(X,Y)$ with the compact open topology is homeomorphic to ...
6
votes
2answers
124 views

Prerequisites for Freedman's proof of the 4-dimensional Poincaré conjecture

I have a good understanding of differential geometry, enough at least to understand many details of Hamilton & Perelman's approach to the 3-dimensional Poincaré conjecture. I have no such ...
-2
votes
1answer
63 views

$H_{n-1}(M;\mathbb{Z})$ is a free abelian group

need help with this problem: show that if $M$ is closed connected oriented n-manifold then $H_{n-1}(M;\mathbb{Z})$ is a free abelian group. thanx.
5
votes
1answer
165 views

Construction of lens spaces

I have a question about the surgery construction of lens spaces. Let $T=S^1 \times D$ be a solid torus. Let $T'$ be another torus. We fix a meridian $m$ and longitude $l$ of the torus. Then the lens ...
0
votes
1answer
39 views

Sufficient conditions for smooth pushout

We restrict ourselves to the category of smooth manifolds and smooth maps. Suppose we have a pair of smooth maps $f:A \to X$ and $g:A \to Y$. A pushout is a pair of smooth maps $p:X \to Z$ and $q:Y ...
10
votes
3answers
399 views

Why care about group actions?

Let X be a space (topological space, manifold, etc) and let the group G act continuously on X. What extra (homotopical, homological, cohomological, diffeomorphical etc) data can extracted from X when ...
0
votes
0answers
30 views

Book of Pullbacks and Pushouts

what books can I consult for properties of pullback and pushouts in algebraic topology? I need to understand the theory of homotopy in algebraic topology and I started to study pullbacks and push ...
0
votes
0answers
48 views

Gluing tori and surgery related lens spaces

I came up with this question when I was thinking about the lens space obtained by an integer surgery along a Hopf link. Let $T_1, T_1', T_2, T_2'$ be a solid torus $S^1 \times D^2$. We regard the ...
4
votes
1answer
89 views

Turn a torus inside out

Let $T=D^2 \times S^1$ be a solid torus, where $D^2$ is a 2-dimensional disk and $S^1$ is a circle. Suppose we have another solid torus $T'$ and we have a homeomorphism $f$ sending a meridian ...
0
votes
0answers
31 views

Intersection form and poincaré duality

Let $ M $ be a $2n$-dimensional compact connected oriented smooth manifold and let $A$, $B$ be two $n$-dimensional submanifolds that intersect transversally. Denote by $A \cdot B$ the sum of the ...
5
votes
3answers
149 views

Applications of Galois theory for topology

Is there any applications of Galois theory in topology? I already have learned Galois theory, and applied it in algebra. Can I get solution of some big problem about topology using Galois theory? ...
5
votes
0answers
79 views

Cohomology of covering space

Let $B$ be a base space and $E$ be a covering space of $B$ what is the relation between $H^2(B,\mathbb{Z})$ and $H^2(E,\mathbb{Z})$.?
8
votes
0answers
71 views

What are necessary and sufficient conditions for the product of spheres to be paralellizable?

Okay, so I found the result that the tangent-bundle of any product of spheres is parallizable, given that some element of the product is either $S^1$, $S^3$, or $S^7$. I prove this as follows, first ...
2
votes
0answers
24 views

uniqueness of Hopf invariant

(Hopf invariant, page 427 of A. Hatcher's Algebrac Topology): Let $f: S^m\longrightarrow S^n$ with $m\geq n$. We can form a CW-complex $C_f$ by attaching a cell $e^{m+1}$ to $S^n$ via $f$. The ...
2
votes
1answer
57 views

Computing the cohomology $H^1(T,\mathbb{Z})$

Let $T$ be a maximal torus of a compact Lie Group. Then how can we compute the first cohomology $H^1(T,\mathbb{Z})$?
2
votes
0answers
47 views

$\mathbb{RP}^2$ does not embed into $\mathbb{R}^3$: reduction to the differentiable case

It is not difficult to see that the real projective plane cannot be embedded into $\mathbb{R}^3$ as a differentiable submanifold (for example one can easily show that the complement would consist of ...
4
votes
0answers
29 views

Second Stiefel-Whitney Class of a Five Manifold

There is a unique rank 4 nontrivial orientable vector bundle over the 2-torus, denote this by $p:E\rightarrow T^2$. Denote the associated sphere bundle by $S(E)$. Then since $S(E)$ is orientable, the ...
2
votes
1answer
34 views

integration of differential forms on covering space

Let $M_1,M_2$ be $n$-dimensional oriented manifolds. Let $f: M_1\longrightarrow M_2$ be an orientation-preserving diffeomorphism. Then for any $\omega\in \Omega^n_c(M_2)$ we have(page 85 of {Madsen: ...
0
votes
0answers
25 views

The piecewise-smooth homotopy

Let $M$ be a smooth manifold. If $\gamma_1:I \rightarrow M$ and $\gamma_2:I\rightarrow M$ are two piecewise-smooth homotopic curves (rel to endpoints), then can we find a map $f:I\times I\rightarrow ...
1
vote
1answer
94 views

a Problem about Degree of Map between Spheres

When I read the book "Algebra Topology-A First Course", I find a problem. It is on the Page 97, Exercise (16.15). Problem We define $f$, $g\colon \mathbb{S}^{n}\to\mathbb{S}^{n}$ to be orthogonal ...
2
votes
1answer
100 views

Are there degree-1 maps from $S^2 \times S^3 \rightarrow S^5$ or from $S^5 \rightarrow S^2\times S^3$?

This is a question from a past qualifying exam I am stuck on: For a smooth map $f:M\rightarrow N$ between smooth, compact, oriented $n$-manifolds, the degree of $f$ is the unique integer $k$ such ...
0
votes
1answer
55 views

$GL(n,\mathbb{C})$ is semi-locally simply connected.

How can we show that $GL(n,\mathbb{C})$ is semi-locally simply connected.
2
votes
2answers
46 views

Loops in $RP^2$

We know that $\pi_1(RP^2)=Z_2.$ How do non-trivial loops in $RP^2$ look like? (If $RP^2$ is the upper hemisphere $D$ with antipodal points of $\partial D$ identified)
3
votes
1answer
52 views

Are space of paths between two different points and space of pointed loops only homotopy equivalent? What about smooth case?

Let $X$ be a path-connected CW-complex and $x$, $y$ points in $X$. Any choice of a path between $x$ and $y$ provides maps (in both directions) between the space $L(x, y)$ of paths from $x$ to $y$ and ...
1
vote
0answers
39 views

Restriction of a line bundle to a a two-cycle

I am reading a paper on Chiral Differential Operators http://arxiv.org/pdf/hep-th/0604179v3.pdf and it says on page 23 that a line bundle over a manifold $C$ can be characterized completely by its ...
5
votes
1answer
47 views

When can a connection be lifted?

Let $P \rightarrow X$ be a principal $G$-bundle, and $P' \rightarrow X'$ be a principal $G'$-bundle. Let $(f',f'')$ be a morphism from $P'$ to $P$, i.e., a pair of maps $f': P' \rightarrow P$ and ...
9
votes
1answer
189 views

Preparing for “differential forms in algebraic topology”?

I'd very much like to read "differential forms in algebraic topology". Apart from background in calculus and linear algbra I've thoroughly went through the first 5 chapters of Munkres. I'm thinking ...
3
votes
0answers
65 views

Concrete non trivial computation of Morse homology

I am studying Morse homology and have found only examples on spheres and tori so far. Of course the homology of these manifolds is better understood by other more standard methods, so I am having ...
0
votes
0answers
37 views

Some questions on definition of Grassman manifolds

In W.M. Boothby, An introduction to differentiable manifolds and Riemannian geometry , Page 64. We use coordinate correspondences $\phi_j : U_j \to R^{k(n-k)}$ in order to define Grassman manifolds ...
3
votes
0answers
42 views

Homological Interpretation of the intersection number

Let $M^r$ and $N^s$ be smooth submanifolds of the smooth manifold $V^{r+s}$ and $M,N,V$ compact, connected and orientable. The Thom-Isomorphism together with the Tubular Neighbourhood Theorem gives me ...
4
votes
3answers
165 views

Application of Hodge decomposition

Hodge decomposition states any $p$ form can be decomposed into three orthogonal $L^2$ components: exact form, co-exact form and hamonic form. But actually we don't know how to decompose a general one. ...
2
votes
1answer
32 views

isometric imbedding of the projective plane

How to isometrically imbed the projective plane (identifying antipodal points of the unit sphere) in $\mathbb{R^5}$? My textbook indicates that there is an isometric imbedding of the projective ...
0
votes
2answers
57 views

Isometry of surfaces in $\mathbb{R}^3$

Let $F$ be an isometry of the Euclidean space $\mathbb{R}^3$. Hence $F$ is orthogonoal transform followed by translation by a constant vector. Let M be a surface of $\mathbb{R}^3$ that is connected, ...
1
vote
0answers
73 views

Orthonormal frame bundle orthogonal to a curve

Let $M$ be a $n$-dimensional smooth riemannian manifold and $\varphi\colon(-\varepsilon,\varepsilon)\rightarrow M$ an embedding. $\varphi$ will denote the image of $\varphi$, too. Consider the bundle ...
2
votes
1answer
52 views

compact surface with two non-intersecting geodesics

I need to find an example of a compact geometric surface M such that Gaussian curvature $K>=0$ M is diffeomorphic to a sphere M has two simply closed geodesics (smoothly closed loops) that ...
3
votes
1answer
76 views

Hopf invariant and the linking number

The Hopf invariant of a map $f:S^{2n-1}\to S^n$ can be defined in various ways, in particular: (1) as the linking number of the preimages of two points and (2) using the cohomology ring of the space ...
6
votes
0answers
284 views

Trying to Understand Lefschetz Pencils

I'm reading on Lefschetz pencils, and I'm trying to understand condition ii) better, tho I would appreciate insights on condition i), and in general. A Lefschetz pencil on a $4$-manifold $X$ is a ...
3
votes
1answer
38 views

Unknotting number formally?

I am reading Colin C. Adams's very nice but not always rigorous "The Knot Book" right now. How does one formalize the unknotting number? (For example, is some restriction on embeddings ...
4
votes
0answers
58 views

Homotopic maps to $S^n$

I'm working through a proof that, given an oriented compact (connected) $n$-manifold $M$ with boundary, any two continuous maps $f,g:M\to S^n$ are homotopic. The proof uses the double of $M$, which is ...
1
vote
2answers
76 views

Frame bundle of orthonormal frames orthogonal to a submanifold.

Suppose we have a smooth manifold $M$ of dimension $m$ with a Riemannian metric and a connected submanifold $N$ of dimension $n$ in $M$ with $n<m-1$. Let $n\le k<m-1$ and consider the bundle ...
1
vote
0answers
54 views

Relative Euler class

In this topic http://mathoverflow.net/questions/93438/euler-class-in-the-non-compact-case you can read about relative Euler class. Can you show me some example of calculation of this class? Do you ...
2
votes
3answers
72 views

Topological spaces with prescribed fundamental groups

The question I am about to ask could have gone to the chat section but I want to have the answers/comments in an easy-to-refer-back-to style. For (connected, pointed) topological spaces with trivial ...