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visits member for 2 years, 2 months
seen Apr 4 at 6:45

Postgrad trying to figure out what to do.


Apr
4
comment On Hatcher's proof that $H_0(X)$ is a direct sum of $\mathbb{Z}$s?
@Mike That's certainly clear! Thanks for your comments.
Apr
4
comment On Hatcher's proof that $H_0(X)$ is a direct sum of $\mathbb{Z}$s?
@Mike Formally, wouldn't set equality of functions require that the domains be equal? Here isn't $[v_0]$ the domain of $\sigma_i$ but $\tau_i|[v_1]$ has domain $[v_1]$? I don't see the domain of $\sigma_0$ specified, only that it maps to $x_0\in X$.
Apr
4
comment On Hatcher's proof that $H_0(X)$ is a direct sum of $\mathbb{Z}$s?
@Mike Thanks. May I ask why does $\sigma_i(v_0)-x_0=\sigma_i-\sigma_0$? Is it because we identify $\sigma_i$ with its image, $\sigma_i(v_0)$, and $\sigma_0$ with its image $x_0$? So these two formal differences are "equal"?
Apr
4
comment On Hatcher's proof that $H_0(X)$ is a direct sum of $\mathbb{Z}$s?
Sorry, I'm confused about what the construction is. First, when viewing $\tau_i\colon I\to X$, this means $\tau_i(0)=x_0$ and $\tau_i(1)=\sigma_i(v_0)$ right? But then Hatcher changes the domain, does this mean now they we identify $v_0$ as the starting point of $I$, and $v_1$ as the end point of $I$, so that $\tau_i|[v_1]=\tau(1)=\sigma_i(v_0)$ and $\tau_i|[v_0]=\tau_i(0)=x_0$? Because I don't see how they coincide.
Apr
4
asked On Hatcher's proof that $H_0(X)$ is a direct sum of $\mathbb{Z}$s?
Apr
4
accepted If $\operatorname{Hom}(X,-)$ and $\operatorname{Hom}(Y,-)$ are isomorphic, why are $X$ and $Y$ isomorphic?
Mar
9
comment If $\operatorname{Hom}(X,-)$ and $\operatorname{Hom}(Y,-)$ are isomorphic, why are $X$ and $Y$ isomorphic?
Thanks! ${}{}{}$
Mar
9
awarded  Critic
Mar
9
asked If $\operatorname{Hom}(X,-)$ and $\operatorname{Hom}(Y,-)$ are isomorphic, why are $X$ and $Y$ isomorphic?
Jan
9
awarded  Nice Question
Nov
29
asked Can $\operatorname{Spec}(R[X])$ ever be finite?
Nov
13
comment If $f\colon X\to S^k$ smooth, $X$ compact and $Z\subseteq S^k$ closed, then $I_2(f,Z)=0$.
Dear Ted, thanks! I'll try to verify these details.
Nov
13
comment If $f\colon X\to S^k$ smooth, $X$ compact and $Z\subseteq S^k$ closed, then $I_2(f,Z)=0$.
@user99680 It's the mod $2$ intersection number, I've put that in now.
Nov
13
revised If $f\colon X\to S^k$ smooth, $X$ compact and $Z\subseteq S^k$ closed, then $I_2(f,Z)=0$.
added 105 characters in body
Nov
13
comment If $f\colon X\to S^k$ smooth, $X$ compact and $Z\subseteq S^k$ closed, then $I_2(f,Z)=0$.
@TedShifrin I think so, Sard's theorem states the set of critical values has measure 0, and a regular value $p$ is one such that the derivative of $f$ is surjective at every point in the fiber of $p$. So I understand that such $p$ exists, it's mostly the stereographic projecting and homotoping that loses me.
Nov
13
asked If $f\colon X\to S^k$ smooth, $X$ compact and $Z\subseteq S^k$ closed, then $I_2(f,Z)=0$.
Aug
19
comment Does $\exp(\ln(I+A))=I+A$ when $\|A\|<1$?
Thanks Alex. What do you mean by "verify the equation $\exp(\log(1+A))=1+A$ is diagonalizable"?
Aug
19
comment Does $\exp(\ln(I+A))=I+A$ when $\|A\|<1$?
Thank you quid.
Aug
19
revised Does $\exp(\ln(I+A))=I+A$ when $\|A\|<1$?
edited title
Aug
19
accepted Why does $\ln(I+A)$ converge when $\|A\|<1$?