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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 …

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 …

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

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 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 $X\times I$ …

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

Still missing in this list, to appear in March 2018, but with some earlier versions online, and very comprehensive: Drutu & Kapovich: „Geometric Group Theory“, http://bookstore.ams.org/coll-63/

This is a Theorem of Yoshida, the reference is Yoshida, Tomoyoshi: ''The η-invariant of hyperbolic 3-manifolds.'' Invent. Math. 81, 473-514 (1985). http://mathlab.snu.ac.kr/~top/articles/Yoshida.pd …

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, …

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 …

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 $M — L$ …

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 …

To answer my own question: the wanted rigidity follows from Theorem 3.4 in http://arxiv.org/pdf/math/9904131.pdf together with the main result from P. Ntolo, Homologie de Leibniz d'algébres de Lie sem …

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 …