Let $\mathcal{A}$ be an additive category. And let $C(\mathcal{A})$ be the category of the chain complexes. Now let $f:X_* \rightarrow Y_*$ be a chain map in $C(\mathcal{A})$ and $C(f)$ be its mapping cone. As we've already known, there's a proposition:     If $C(f)$ is contractible, then $f:X_* \rightarrow Y_*$ is a homotopy equivalence.   My question: Is the inverse problem true? To be explicitly, if $f:X_* \rightarrow Y_*$ is a homotopy equivalence, then does $C(f)$ necessarily to be contractible? I guess that this is not true in general for a part of the proof of the proposition doesn't holds conversely. Am I right? If I'm right, is there any counterexample? Thanks in advance.
Let $f: K^\bullet \to L^\bullet$ be homotopy equivalence. From theory of triangulated categories it follows that $C(f)^\bullet$ is contractible. But how can i produce explicit formula for homotopy of $Id_{C(f)^\bullet}$ and $0_{C(f)^\bullet}$?
Could the universe's antimatter be hiding in black holes? As far as we know almost all the mass in the universe is matter, not antimatter. There are three scenarios:   The universe started off with more matter than antimatter and the small amount of antimatter has mostly been converted to energy via collisions with matter The universe started off with equal amounts of matter and antimatter and some unknown process caused the situation we see today The universe started off with less matter than antimatter and somehow the ratio reversed (I've never heard this discussed as it seems so unlikely)   We reason that mass is matter because if there were large amounts of antimatter they should produce antimatter galaxies, which would give off huge amounts of energy when they interact with matter galaxies. This has never been observed so we conclude that all galaxies, therefore all mass in the universe is matter.  As I understand it there are two main types of black holes, stellar remnants (the collapsed remnants of dead stars with more than two solar masses) and super-massive black holes at the centre of many galaxies, including one thought to be at the centre of our own galaxy.  Stellar black holes are obviously originally composed of matter, for the same reason that all galaxies are composed of matter.  We don't know how super-massive black holes formed. Is there enough mass in them to contain all the antimatter in the universe if matter and antimatter were originally created in equal quantities? I realise this shifts the unknown mechanism from "why was more matter created than antimatter" to "why is antimatter more likely to form super-massive black holes" but is this hypothetical scenario possible? How to add \part only to TOC? Is my baby's gender an example of Schrodinger's cat? Inverse Laplace transform of $\frac{1}{s\sinh{c\sqrt{s}}}$ I have some questions.  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