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A closed surface of genus g can be cut along 2g closed curves (all at one base point) to obtain a 4g-gon. Do this for both surfaces. Then choose a homeomorphism between the 4g-gons which matches the c …

The quotes are from Thurston's survey paper Three dimensional manifolds, kleinian groups and hyperbolic geometry page 362: 2.6. THEOREM [Th 1]. Suppose $L \subset M^3$ is a link such that $M — L$ …

First I think, the covering $T^2\to S^2$ has deck group $Z/2Z\oplus Z/2Z$ generated by $(x,y)\to (-x,y)$ and $(x,y)\to (x,-y)$. Then for the 3-dimensional case, you consider the action of $$Z/2Z\oplu …

(Edit: Everything what follows is about closed and orientable $3$-manifolds.) Non-spherical geometric $3$-manifolds are determined by their fundamental group. This is proved in Peter Scott's paper "T …

Everything you would like to know about Fake lens spaces is on http://www.map.mpim-bonn.mpg.de/Fake_lens_spaces by Tibor Macko. A Fake lens space is the Quotient of a sphere by a cyclic group acting …

This is completely elementary. From the H-space structure $X\times X\to X$ you construct the Hopf fibration $X*X\to SX$ via $(x,t,y)\to (xy,t)$, where you think of $SX$ as the quotient of $X\times I$ …

Foliations of surfaces only exist on surfaces of Euler characteristic 0 and they are all built from suspension foliations and Reeb components. (See the book by Hector and Hirsch). On 3-manifolds ther …

This follows from Theorem 2 in Thurston's 1974 paper "A generalization of the Reeb stability theorem", at least if $H^1(L,R)=0$. https://core.ac.uk/download/pdf/82172971.pdf

If you define Lie algebra cohomology via injective resolutions, then your claim is trivially true, because you can take the injective resolution which is I in degree 0 and 0 in higher degrees. To see …

The figure eight knot complement is a fiber bundle over the circle with fiber a once-punctured torus. There is a very natural and easy construction of ideal triangulations (and hence hyperbolic stru …

Hitchin 1987 extended the work of Atiyah-Bott to study the topology of the moduli space of Higgs bundles on $\Sigma$ via the Yang-Mills functional. Simpson 1988 proved that this moduli space agrees wi …

The LS-category of a connected, second countable, n-dimensional manifold satisfies $$cat(X)\le n+1.$$ I suppose this is well known, you find a proof in http://math.ucr.edu/~res/math246A/cuplength.pdf

The reference for the first appearance of the conjecture (still without the condition that the PD group has to be a priori finitely presented) seems to be http://www.worldcat.org/title/homological-gr …

Take an n-sphere with its standard metric of positive curvature, cut out an n-ball and reglue it by a diffeomorphism of the bounding n-1-sphere. If the diffeomorphism is not isotopic to the identity, …

I remember some talks some time ago about proofs of nonexistence of Lagrangian Kleinian bottles in ${\mathbb C}^2$ for the standard symplectic structure, mentioning that this were the only compact sur …