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        The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.

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        Problem about the homology groups of the complex projective space $\mathbb{C}P_n$ [on hold]

        My question is how we can compute the homology groups of the complex projective space $\mathbb{C}P_n$ by the following Corollary5.4 at page 31 in Milnor's book: Corollary5.4 If $c_{\lambda+1}=c_{\...
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        Is it necessary for a conservative vector field to be $C^1$?

        Is it necessary for a conservative vector field on a domain $A \subset \mathbb{R}^3$ to be $C^1$? I know that if a vector field $F$ is defined on a simply connected domain $A$ simply connected and $...
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        Diffeomorphism type of Ricci-flat four manifolds

        Let $(M,g)$ be an irreducible compact and simply connected Ricci-flat Riemannian four-manifold. My first questions are as follows: A) Is there a classification of the possible homeomorphism types of ...
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        Is there a vector field such that one differential form is the Lie derivative of the other?

        I'm looking for a reference or answer for the following question: Let $M$ be an (compact and orientable, if it helps) smooth manifold and $\nu$ and $\mu$ two differential forms. I'm looking for ...
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        Dimensions of the instanton moduli space from Atiyah-Hitchin-Singer

        Atiyah-Hitchin-Singer Ref 1 states that the number of virtual dimensions of the instanton moduli space for SU(N) Yang-Mills theory with topological charge $\mathcal{Q}$ over a manifold $X$ is given ...
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        183 views

        Embedding problem for 3-manifolds attacked via 4-manifolds

        In this archiv paper which is continuation of following: Borodzik, Maciej; Némethi, András; Ranicki, Andrew, Morse theory for manifolds with boundary, Algebr. Geom. Topol. 16, No. 2, 971-1023 (2016). ...
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        The limitation of $G$ and loop group $\Omega G$ in Atiyah's and Donaldson's work on Instantons

        In Atiyah's work [Ref. 1], Atiyah states that "Essentially we shall show (at least for $G$ a classical group and probably for all $G$) that Yang-Mills instantons in 4D can be naturally identified with ...
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        120 views

        A geometric property about certain polynomials in two variables

        Assume that $p(x,y)$ is a polynomial in $\mathbb{R}[x, y]$ in the form $$ p=p_{2n}+ p_{2n-1}+\ldots +p_1+p_0$$ where $p_i$ is a homogenous polynomial of degree $i$. Moreover we assume that the last ...
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        156 views

        Do an unlinked trefoil and figure-eight cobound an annulus in $B^4$?

        Let $K_1$ the trefoil (left or right hopefully does not matter?) and let $K_2$ be the figure-eight knot in $S^3 = \partial B^4$. Are there any smooth properly embedded annulus $A$ in $B^4$ with $\...
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        Which 3-manifolds are known to admit exotic pairs of bounding 4-manifolds?

        Let $M$ be a compact connected three manifold. By an exotic pair of bounding 4-manifolds, I mean two smooth 4-manifolds $X_1,X_2$ such that $X_1$ and $X_2$ are homeomorphic but not diffeomorphic, and ...
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        88 views

        Minimum number of double points over all immersed disks

        Let $K$ be a knot in the boundary of a compact smooth 4-manifold $X$, and suppose that $K$ is the the kernel of $\pi_1(\partial X) \to \pi_1(X)$. Then $K$ is the boundary of some immersed disk $D \to ...
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        143 views

        Smooth functions on subsets of $\mathbb{R}^n$

        I am teaching a course in basic differential topology, and, following e.g. Milnor, I defined functions of class $C^k$ on subsets of the Euclidean space $\mathbb{R}^n$ as follows. Let $f\colon X\to \...
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        2answers
        296 views

        Nash isometric embedding theorem with keeping the symplectic structures of our ambient spaces

        I apologize in advance if this question has an obvious answer. Let $(M,g)$ be a Riemannian manifold. Then the tangent bundle $TM$ carries a natural symplectic structure $\omega_g$. In fact $\omega_g$...
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        386 views

        Measures and differential forms on manifolds

        Let $M$ be a differentiable manifold. Let $\mu$ be a (probability) measure on $M$. What are the conditions under which $\mu$ is given by a differential form on $M$? I imagine some sort of ...
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        103 views

        Theorems similar to Tischler fibering theorem

        Tischler theorem states that the existence of a nowhere vanishing closed $1$-form in a compact manifold $M$ implies that the manifold fibers over $S^1$. Do you know any other diffential topology ...

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